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## Statistics and probability

Course: statistics and probability > unit 12, simple hypothesis testing.

- Idea behind hypothesis testing
- Examples of null and alternative hypotheses
- Writing null and alternative hypotheses
- P-values and significance tests
- Comparing P-values to different significance levels
- Estimating a P-value from a simulation
- Estimating P-values from simulations
- Using P-values to make conclusions
- Your answer should be
- an integer, like 6
- an exact decimal, like 0.75
- a simplified proper fraction, like 3 / 5
- a simplified improper fraction, like 7 / 4
- a mixed number, like 1 3 / 4
- a percent, like 12.34 %
- (Choice A) We cannot reject the hypothesis. A We cannot reject the hypothesis.
- (Choice B) We should reject the hypothesis. B We should reject the hypothesis.

## with Answer, Solution | Statistical Inference - Hypothesis Testing: Solved Example Problems | 12th Business Maths and Statistics : Chapter 8 : Sampling Techniques and Statistical Inference

Chapter: 12th business maths and statistics : chapter 8 : sampling techniques and statistical inference, hypothesis testing: solved example problems.

Example 8.14

An auto company decided to introduce a new six cylinder car whose mean petrol consumption is claimed to be lower than that of the existing auto engine. It was found that the mean petrol consumption for the 50 cars was 10 km per litre with a standard deviation of 3.5 km per litre. Test at 5% level of significance, whether the claim of the new car petrol consumption is 9.5 km per litre on the average is acceptable.

Population mean μ = 9.5 km

Since population SD is unknown we consider σ = s

The sample is a large sample and so we apply Z-test

Null Hypothesis: There is no significant difference between the sample average and the company’s claim, i.e., H 0 : μ = 9.5

Alternative Hypothesis: There is significant difference between the sample average and the company’s claim, i.e., H 1 : μ ≠ 9.5 (two tailed test)

The level of significance α = 5% = 0.05

Applying the test statistic

Thus the calculated value 1.01 and the significant value or table value Z α /2 = 1.96

Comparing the calculated and table value ,Here Z < Z α /2 i.e., 1.01<1.96.

Inference:Since the calculated value is less than table value i.e., Z < Z α /2 at 5% level of sinificance, the null hypothesis H 0 is accepted. Hence we conclude that the company’s claim that the new car petrol consumption is 9.5 km per litre is acceptable.

Example 8.15

A manufacturer of ball pens claims that a certain pen he manufactures has a mean writing life of 400 pages with a standard deviation of 20 pages. A purchasing agent selects a sample of 100 pens and puts them for test. The mean writing life for the sample was 390 pages. Should the purchasing agent reject the manufactures claim at 1% level?

Population SD σ = 20 pages

The sample is a large sample and so we apply Z -test

Null Hypothesis: There is no significant difference between the sample mean and the population mean of writing life of pen he manufactures, i.e., H 0 : μ = 400

Alternative Hypothesis: There is significant difference between the sample mean and the population mean of writing life of pen he manufactures, i.e., H 1 : μ ≠ 400 (two tailed test)

The level of significance a = 1% = 0.01

Thus the calculated value |Z| = 5 and the significant value or table value Z α /2 = 2.58

Comparing the calculated and table values, we found Z > Z α /2 i.e., 5 > 2.58

Inference: Since the calculated value is greater than table value i.e., Z > Z α /2 at 1% level of significance, the null hypothesis is rejected and Therefore we concluded that μ ≠ 400 and the manufacturer’s claim is rejected at 1% level of significance.

Example 8.16

(i) A sample of 900 members has a mean 3.4 cm and SD 2.61 cm. Is the sample taken from a large population with mean 3.25 cm. and SD 2.62 cm?

(ii) If the population is normal and its mean is unknown, find the 95% and 98% confidence limits of true mean.

Population mean μ= 3.25 cm, Population SD σ = 2.61 cm

Null Hypothesis H 0 : μ = 3.25 cm (the sample has been drawn from the population mean

μ = 3.25 cm and SD σ = 2.61 cm)

Alternative Hypothesis H 1 : μ ≠ 3.25 cm (two tail) i.e., the sample has not been drawn from the population mean μ = 3.25 cm and SD σ = 2.61 cm.

Teststatistic:

∴ Z = 1.724

Thus the calculated and the significant value or table value Z α /2 = 1.96

Comparing the calculated and table values, Z < Z α /2 i.e., 1.724 < 1.96

Inference:Since the calculated value is less than table value i.e., Z > Z α /2 at 5% level of significance, the null hypothesis is accepted. Hence we conclude that the2 data doesn’t provide us any evidence against the null hypothesis. Therefore, the sample has been drawn from the population mean μ = 3.25 cm and SD, σ = 2.61 cm.

(ii) Confidence limits

95% confidential limits for the population mean μ are :

3.4− (1.96× 0.087)≤ μ ≤ 3.4+ (1.96× 0.087)

3.229≤ μ ≤ 3.571

98% confidential limits for the population mean are :

3.4− (2.33× 0.087)≤ μ ≤ 3.4+ (2.33× 0.087)

3.197 ≤ μ≤ 3.603

Therefore,95% confidential limits is (3.229,3.571) and 98% confidential limits is (3.197,3.603).

Example 8.17

The mean weekly sales of soap bars in departmental stores were 146.3 bars per store. After an advertising campaign the mean weekly sales in 400 stores for a typical week increased to 153.7 and showed a standard deviation of 17.2. Was the advertising campaign successful?

Sample size n = 400 stores

Sample SD s = 17.2 bars

Population mean μ = 146.3 bars

Since population SD is unknown we can consider the sample SD s = σ

Null Hypothesis. The advertising campaign is not successful i.e, H 0 : μ = 146.3 (There is no significant difference between the mean weekly sales of soap bars in department stores before and after advertising campaign)

Alternative Hypothesis H 1 : μ > 143.3 (Right tail test). The advertising campaign was successful

Level of significance a = 0.05

Test statistic

∴ Z = 8.605

Comparing the calculated value Z = 8.605 and the significant value or table value Z α = 1.645 . we get 8.605 > 1.645

Inference: Since, the calculated value is much greater than table value i.e., Z > Z α , it is highly significant at 5% level of significance. Hence we reject the null hypothesis H0 and conclude that the advertising campaign was definitely successful in promoting sales.

Example 8.18

The wages of the factory workers are assumed to be normally distributed with mean and variance 25. A random sample of 50 workers gives the total wages equal to ₹ 2,550. Test the hypothesis μ = 52, against the alternative hypothesis μ = 49 at 1% level of significance.

Sample size n = 50 workers

Since alternative hypothesis is of two tailed test we can take | Z | = 1.4142

Critical value at 1% level of significance is Z α /2 = 2.58

Inference: Since the calculated value is less than table value i.e., Z < Za at 1% level of significance, the null hypothesis H 0 is accepted. Therefore, we conclude 2that there is no significant difference between the sample mean and population mean μ= 52 and SD σ = 5.

Example 8.19

An ambulance service claims that it takes on the average 8.9 minutes to reach its destination in emergency calls. To check on this claim, the agency which licenses ambulance services has them timed on 50 emergency calls, getting a mean of 9.3 minutes with a standard deviation of 1.6 minutes. What can they conclude at the level of significance.

Calculated value Z = 1.7676

Critical value at 5% level of significance is Z α /2 = 1.96

Inference: Since the calculated value is less than table value i.e., Z < Z α /2 at 5% level of significance, the null hypothesis is accepted. Therefore we conclude that an ambulance service claims on the average 8.9 minutes to reach its destination in emergency calls.

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- Null and Alternative Hypotheses | Definitions & Examples

## Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

- Null hypothesis ( H 0 ): There’s no effect in the population .
- Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

## Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:

- The null hypothesis ( H 0 ) answers “No, there’s no effect in the population.”
- The alternative hypothesis ( H a ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

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The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.

## Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

## Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

- They’re both answers to the research question.
- They both make claims about the population.
- They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

## General template sentences

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

- Null hypothesis ( H 0 ): Independent variable does not affect dependent variable.
- Alternative hypothesis ( H a ): Independent variable affects dependent variable.

## Test-specific template sentences

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

- Normal distribution
- Descriptive statistics
- Measures of central tendency
- Correlation coefficient

Methodology

- Cluster sampling
- Stratified sampling
- Types of interviews
- Cohort study
- Thematic analysis

Research bias

- Implicit bias
- Cognitive bias
- Survivorship bias
- Availability heuristic
- Nonresponse bias
- Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

## Cite this Scribbr article

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## 9.2: Hypothesis Testing

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All hypotheses tests have the same basic steps:

- Determine the hypothesis : What are we trying to figure out? This is formally written as the null and alternative hypotheses.
- Calculate the evidence : This will be a test statistics and either a critical value or a p-value.
- Make a decision : The options will be Reject the Null Hypothesis or Do not Reject the Null Hypothesis.
- Determine the conclusion : What does the decision mean in terms of the problem given?

## Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

\(H_0\): The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

\(H_a\): The alternative hypothesis: It is a claim about the population that is contradictory to \(H_0\) and what we conclude when we reject \(H_0\). This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject \(H_0\)" if the sample information favors the alternative hypothesis or "do not reject \(H_0\)" or "decline to reject \(H_0\)" if the sample information is insufficient to reject the null hypothesis.

\(H_{0}\) always has a symbol with an equal in it. \(H_{a}\) never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

## Example \(\PageIndex{1}\)

- \(H_{0}\): No more than 30% of the registered voters in Santa Clara County voted in the primary election. \(p \leq 30\)
- \(H_{a}\): More than 30% of the registered voters in Santa Clara County voted in the primary election. \(p > 30\)

## Exercise \(\PageIndex{1}\)

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

- \(H_{0}\): The drug reduces cholesterol by 25%. \(p = 0.25\)
- \(H_{a}\): The drug does not reduce cholesterol by 25%. \(p \neq 0.25\)

## Example \(\PageIndex{2}\)

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

- \(H_{0}: \mu = 2.0\)
- \(H_{a}: \mu \neq 2.0\)

## Exercise \(\PageIndex{2}\)

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.

- \(H_{0}: \mu_ 66\)
- \(H_{a}: \mu_ 66\)
- \(H_{0}: \mu = 66\)
- \(H_{a}: \mu \neq 66\)

## Example \(\PageIndex{3}\)

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

- \(H_{0}: \mu \geq 66\)
- \(H_{a}: \mu < 66\)

## Exercise \(\PageIndex{3}\)

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

- \(H_{0}: \mu_ 45\)
- \(H_{a}: \mu_ 45\)
- \(H_{0}: \mu \geq 45\)
- \(H_{a}: \mu < 45\)

## Example \(\PageIndex{4}\)

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

- \(H_{0}: p \leq 0.066\)
- \(H_{a}: p > 0.066\)

## Exercise \(\PageIndex{4}\)

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (\(=, \neq, \geq, <, \leq, >\)) for the null and alternative hypotheses.

- \(H_{0}: p_ 0.40\)
- \(H_{a}: p_ 0.40\)
- \(H_{0}: p = 0.40\)
- \(H_{a}: p > 0.40\)

## COLLABORATIVE EXERCISE

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

## Outcomes and the Type I and Type II Errors

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis \(H_{0}\) and the decision to reject or not. The outcomes are summarized in the following table:

The four possible outcomes in the table are:

- The decision is not to reject \(H_{0}\) when \(H_{0}\) is true (correct decision).
- The decision is to reject \(H_{0}\) when \(H_{0}\) is true (incorrect decision known as aType I error).
- The decision is not to reject \(H_{0}\) when, in fact, \(H_{0}\) is false (incorrect decision known as a Type II error).
- The decision is to reject \(H_{0}\) when \(H_{0}\) is false ( correct decision whose probability is called the Power of the Test ).

Each of the errors occurs with a particular probability. The Greek letters \(\alpha\) and \(\beta\) represent the probabilities.

- \(\alpha =\) probability of a Type I error \(= P(\text{Type I error}) =\) probability of rejecting the null hypothesis when the null hypothesis is true.
- \(\beta =\) probability of a Type II error \(= P(\text{Type II error}) =\) probability of not rejecting the null hypothesis when the null hypothesis is false.

\(\alpha\) and \(\beta\) should be as small as possible because they are probabilities of errors. They are rarely zero.

The Power of the Test is \(1 - \beta\). Ideally, we want a high power that is as close to one as possible. Increasing the sample size can increase the Power of the Test. The following are examples of Type I and Type II errors.

## Example \(\PageIndex{5}\): Type I vs. Type II errors

Suppose the null hypothesis, \(H_{0}\), is: Frank's rock climbing equipment is safe.

- Type I error : Frank thinks that his rock climbing equipment may not be safe when, in fact, it really is safe.
- Type II error : Frank thinks that his rock climbing equipment may be safe when, in fact, it is not safe.

\(\alpha =\) probability that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is safe.

\(\beta =\) probability that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe.

Notice that, in this case, the error with the greater consequence is the Type II error. (If Frank thinks his rock climbing equipment is safe, he will go ahead and use it.)

## Exercise \(\PageIndex{5}\)

Suppose the null hypothesis, \(H_{0}\), is: the blood cultures contain no traces of pathogen \(X\). State the Type I and Type II errors.

- Type I error : The researcher thinks the blood cultures do contain traces of pathogen \(X\), when in fact, they do not.
- Type II error : The researcher thinks the blood cultures do not contain traces of pathogen \(X\), when in fact, they do.

## Example \(\PageIndex{6}\)

Suppose the null hypothesis, \(H_{0}\), is: The victim of an automobile accident is alive when he arrives at the emergency room of a hospital.

- Type I error : The emergency crew thinks that the victim is dead when, in fact, the victim is alive.
- Type II error : The emergency crew does not know if the victim is alive when, in fact, the victim is dead.

\(\alpha =\) probability that the emergency crew thinks the victim is dead when, in fact, he is really alive \(= P(\text{Type I error})\).

\(\beta =\) probability that the emergency crew does not know if the victim is alive when, in fact, the victim is dead \(= P(\text{Type II error})\).

The error with the greater consequence is the Type I error. (If the emergency crew thinks the victim is dead, they will not treat him.)

## Exercise \(\PageIndex{6}\)

Suppose the null hypothesis, \(H_{0}\), is: a patient is not sick. Which type of error has the greater consequence, Type I or Type II?

The error with the greater consequence is the Type II error: the patient will be thought well when, in fact, he is sick, so he will not get treatment.

## Example \(\PageIndex{7}\)

It’s a Boy Genetic Labs claim to be able to increase the likelihood that a pregnancy will result in a boy being born. Statisticians want to test the claim. Suppose that the null hypothesis, \(H_{0}\), is: It’s a Boy Genetic Labs has no effect on gender outcome.

- Type I error : This results when a true null hypothesis is rejected. In the context of this scenario, we would state that we believe that It’s a Boy Genetic Labs influences the gender outcome, when in fact it has no effect. The probability of this error occurring is denoted by the Greek letter alpha, \(\alpha\).
- Type II error : This results when we fail to reject a false null hypothesis. In context, we would state that It’s a Boy Genetic Labs does not influence the gender outcome of a pregnancy when, in fact, it does. The probability of this error occurring is denoted by the Greek letter beta, \(\beta\).

The error of greater consequence would be the Type I error since couples would use the It’s a Boy Genetic Labs product in hopes of increasing the chances of having a boy.

## Exercise \(\PageIndex{7}\)

“Red tide” is a bloom of poison-producing algae–a few different species of a class of plankton called dinoflagellates. When the weather and water conditions cause these blooms, shellfish such as clams living in the area develop dangerous levels of a paralysis-inducing toxin. In Massachusetts, the Division of Marine Fisheries (DMF) monitors levels of the toxin in shellfish by regular sampling of shellfish along the coastline. If the mean level of toxin in clams exceeds 800 μg (micrograms) of toxin per kg of clam meat in any area, clam harvesting is banned there until the bloom is over and levels of toxin in clams subside. Describe both a Type I and a Type II error in this context, and state which error has the greater consequence.

In this scenario, an appropriate null hypothesis would be \(H_{0}\): the mean level of toxins is at most \(800 \mu\text{g}\), \(H_{0}: \mu_{0} \leq 800 \mu\text{g}\).

## Example \(\PageIndex{8}\)

A certain experimental drug claims a cure rate of at least 75% for males with prostate cancer. Describe both the Type I and Type II errors in context. Which error is the more serious?

- Type I : A cancer patient believes the cure rate for the drug is less than 75% when it actually is at least 75%.
- Type II : A cancer patient believes the experimental drug has at least a 75% cure rate when it has a cure rate that is less than 75%.

In this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75% of the time, this most likely will influence the patient’s (and doctor’s) choice about whether to use the drug as a treatment option.

## Exercise \(\PageIndex{8}\)

Determine both Type I and Type II errors for the following scenario:

Assume a null hypothesis, \(H_{0}\), that states the percentage of adults with jobs is at least 88%. Identify the Type I and Type II errors from these four statements.

- Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%
- Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.
- Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.
- Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%.

Type I error: c

Type I error: b

## Distribution Needed for Hypothesis Testing

Earlier in the course, we discussed sampling distributions. Particular distributions are associated with hypothesis testing. Perform tests of a population mean using a normal distribution or a Student's \(t\)-distribution. (Remember, use a Student's \(t\)-distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal.) We perform tests of a population proportion using a normal distribution (usually \(n\) is large or the sample size is large).

If you are testing a single population mean, the distribution for the test is for means :

\[\bar{X} - N\left(\mu_{x}, \frac{\sigma_{x}}{\sqrt{n}}\right)\]

The population parameter is \(\mu\). The estimated value (point estimate) for \(\mu\) is \(\bar{x}\), the sample mean.

If you are testing a single population proportion, the distribution for the test is for proportions or percentages:

\[P' - N\left(p, \sqrt{\frac{p-q}{n}}\right)\]

The population parameter is \(p\). The estimated value (point estimate) for \(p\) is \(p′\). \(p' = \frac{x}{n}\) where \(x\) is the number of successes and n is the sample size.

## Assumptions

When you perform a hypothesis test of a single population mean \(\mu\) using a Student's \(t\)-distribution (often called a \(t\)-test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation. (Note that if the sample size is sufficiently large, a \(t\)-test will work even if the population is not approximately normally distributed).

When you perform a hypothesis test of a single population mean \(\mu\) using a normal distribution (often called a \(z\)-test), you take a simple random sample from the population. The population you are testing is normally distributed or your sample size is sufficiently large. You know the value of the population standard deviation which, in reality, is rarely known.

When you perform a hypothesis test of a single population proportion \(p\), you take a simple random sample from the population. You must meet the conditions for a binomial distribution which are: there are a certain number \(n\) of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success \(p\). The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities \(np\) and \(nq\) must both be greater than five \((np > 5\) and \(nq > 5)\). Then the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with \(\mu = p\) and \(\sigma = \sqrt{\frac{pq}{n}}\). Remember that \(q = 1 – p\).

## Rare Events, the Sample, Decision and Conclusion

Establishing the type of distribution, sample size, and known or unknown standard deviation can help you figure out how to go about a hypothesis test. However, there are several other factors you should consider when working out a hypothesis test.

## Rare Events

Suppose you make an assumption about a property of the population (this assumption is the null hypothesis). Then you gather sample data randomly. If the sample has properties that would be very unlikely to occur if the assumption is true, then you would conclude that your assumption about the population is probably incorrect. (Remember that your assumption is just an assumption—it is not a fact and it may or may not be true. But your sample data are real and the data are showing you a fact that seems to contradict your assumption.)

For example, Didi and Ali are at a birthday party of a very wealthy friend. They hurry to be first in line to grab a prize from a tall basket that they cannot see inside because they will be blindfolded. There are 200 plastic bubbles in the basket and Didi and Ali have been told that there is only one with a $100 bill. Didi is the first person to reach into the basket and pull out a bubble. Her bubble contains a $100 bill. The probability of this happening is \(\frac{1}{200} = 0.005\). Because this is so unlikely, Ali is hoping that what the two of them were told is wrong and there are more $100 bills in the basket. A "rare event" has occurred (Didi getting the $100 bill) so Ali doubts the assumption about only one $100 bill being in the basket.

## Using the Sample to Test the Null Hypothesis

Use the sample data to calculate the actual probability of getting the test result, called the \(p\)-value. The \(p\)-value is the probability that, if the null hypothesis is true, the results from another randomly selected sample will be as extreme or more extreme as the results obtained from the given sample.

A large \(p\)-value calculated from the data indicates that we should not reject the null hypothesis. The smaller the \(p\)-value, the more unlikely the outcome, and the stronger the evidence is against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against it.

Draw a graph that shows the \(p\)-value. The hypothesis test is easier to perform if you use a graph because you see the problem more clearly.

## Example \(\PageIndex{9}\)

Suppose a baker claims that his bread height is more than 15 cm, on average. Several of his customers do not believe him. To persuade his customers that he is right, the baker decides to do a hypothesis test. He bakes 10 loaves of bread. The mean height of the sample loaves is 17 cm. The baker knows from baking hundreds of loaves of bread that the standard deviation for the height is 0.5 cm. and the distribution of heights is normal.

- The null hypothesis could be \(H_{0}: \mu \leq 15\)
- The alternate hypothesis is \(H_{a}: \mu > 15\)

The words "is more than" translates as a "\(>\)" so "\(\mu > 15\)" goes into the alternate hypothesis. The null hypothesis must contradict the alternate hypothesis.

Since \(\sigma\) is known (\(\sigma = 0.5 cm.\)), the distribution for the population is known to be normal with mean \(μ = 15\) and standard deviation

\[\dfrac{\sigma}{\sqrt{n}} = \frac{0.5}{\sqrt{10}} = 0.16. \nonumber\]

Suppose the null hypothesis is true (the mean height of the loaves is no more than 15 cm). Then is the mean height (17 cm) calculated from the sample unexpectedly large? The hypothesis test works by asking the question how unlikely the sample mean would be if the null hypothesis were true. The graph shows how far out the sample mean is on the normal curve. The p -value is the probability that, if we were to take other samples, any other sample mean would fall at least as far out as 17 cm.

The \(p\) -value, then, is the probability that a sample mean is the same or greater than 17 cm. when the population mean is, in fact, 15 cm. We can calculate this probability using the normal distribution for means.

\(p\text{-value} = P(\bar{x} > 17)\) which is approximately zero.

A \(p\)-value of approximately zero tells us that it is highly unlikely that a loaf of bread rises no more than 15 cm, on average. That is, almost 0% of all loaves of bread would be at least as high as 17 cm. purely by CHANCE had the population mean height really been 15 cm. Because the outcome of 17 cm. is so unlikely (meaning it is happening NOT by chance alone) , we conclude that the evidence is strongly against the null hypothesis (the mean height is at most 15 cm.). There is sufficient evidence that the true mean height for the population of the baker's loaves of bread is greater than 15 cm.

## Exercise \(\PageIndex{9}\)

A normal distribution has a standard deviation of 1. We want to verify a claim that the mean is greater than 12. A sample of 36 is taken with a sample mean of 12.5.

- \(H_{0}: \mu leq 12\)
- \(H_{a}: \mu > 12\)

The \(p\)-value is 0.0013

Draw a graph that shows the \(p\)-value.

\(p\text{-value} = 0.0013\)

## Decision and Conclusion

A systematic way to make a decision of whether to reject or not reject the null hypothesis is to compare the \(p\)-value and a preset or preconceived \(\alpha\) (also called a " significance level "). A preset \(\alpha\) is the probability of a Type I error (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem.

When you make a decision to reject or not reject \(H_{0}\), do as follows:

- If \(\alpha > p\text{-value}\), reject \(H_{0}\). The results of the sample data are significant. There is sufficient evidence to conclude that \(H_{0}\) is an incorrect belief and that the alternative hypothesis, \(H_{a}\), may be correct.
- If \(\alpha \leq p\text{-value}\), do not reject \(H_{0}\). The results of the sample data are not significant.There is not sufficient evidence to conclude that the alternative hypothesis,\(H_{a}\), may be correct.

When you "do not reject \(H_{0}\)", it does not mean that you should believe that H 0 is true. It simply means that the sample data have failed to provide sufficient evidence to cast serious doubt about the truthfulness of \(H_{0}\).

Conclusion: After you make your decision, write a thoughtful conclusion about the hypotheses in terms of the given problem.

## Example \(\PageIndex{10}\)

When using the \(p\)-value to evaluate a hypothesis test, it is sometimes useful to use the following memory device

- If the \(p\)-value is low, the null must go.
- If the \(p\)-value is high, the null must fly.

This memory aid relates a \(p\)-value less than the established alpha (the \(p\) is low) as rejecting the null hypothesis and, likewise, relates a \(p\)-value higher than the established alpha (the \(p\) is high) as not rejecting the null hypothesis.

Fill in the blanks.

Reject the null hypothesis when ______________________________________.

The results of the sample data _____________________________________.

Do not reject the null when hypothesis when __________________________________________.

The results of the sample data ____________________________________________.

Reject the null hypothesis when the \(p\) -value is less than the established alpha value . The results of the sample data support the alternative hypothesis .

Do not reject the null hypothesis when the \(p\) -value is greater than the established alpha value . The results of the sample data do not support the alternative hypothesis .

## Exercise \(\PageIndex{10}\)

It’s a Boy Genetics Labs claim their procedures improve the chances of a boy being born. The results for a test of a single population proportion are as follows:

- \(H_{0}: p = 0.50, H_{a}: p > 0.50\)
- \(\alpha = 0.01\)
- \(p\text{-value} = 0.025\)

Interpret the results and state a conclusion in simple, non-technical terms.

Since the \(p\)-value is greater than the established alpha value (the \(p\)-value is high), we do not reject the null hypothesis. There is not enough evidence to support It’s a Boy Genetics Labs' stated claim that their procedures improve the chances of a boy being born.

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:

- Evaluate the null hypothesis , typically denoted with \(H_{0}\). The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality \((=, \leq \text{or} \geq)\)
- Always write the alternative hypothesis , typically denoted with \(H_{a}\) or \(H_{1}\), using less than, greater than, or not equals symbols, i.e., \((\neq, >, \text{or} <)\).
- If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
- Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected. The probabilities of these errors are denoted by the Greek letters \(\alpha\) and \(\beta\), for a Type I and a Type II error respectively. The power of the test, \(1 - \beta\), quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable.

In order for a hypothesis test’s results to be generalized to a population, certain requirements must be satisfied.

When testing for a single population mean:

- A Student's \(t\)-test should be used if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation.
- The normal test will work if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with a known standard deviation.

When testing a single population proportion use a normal test for a single population proportion if the data comes from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of successes and the mean number of failures satisfy the conditions: \(np > 5\) and \(nq > 5\) where \(n\) is the sample size, \(p\) is the probability of a success, and \(q\) is the probability of a failure.

When the probability of an event occurring is low, and it happens, it is called a rare event. Rare events are important to consider in hypothesis testing because they can inform your willingness not to reject or to reject a null hypothesis. To test a null hypothesis, find the p -value for the sample data and graph the results. When deciding whether or not to reject the null the hypothesis, keep these two parameters in mind:

- \(\alpha > p-value\), reject the null hypothesis
- \(\alpha \leq p-value\), do not reject the null hypothesis

## Formula Review

\(H_{0}\) and \(H_{a}\) are contradictory.

- If \(\alpha \leq p\)-value, then do not reject \(H_{0}\).
- If\(\alpha > p\)-value, then reject \(H_{0}\).

\(\alpha\) is preconceived. Its value is set before the hypothesis test starts. The \(p\)-value is calculated from the data.

If there is no given preconceived \(\alpha\), then use \(\alpha = 0.05\).

Types of Hypothesis Tests

- Single population mean, known population variance (or standard deviation): Normal test .
- Single population mean, unknown population variance (or standard deviation): Student's \(t\)-test .
- Single population proportion: Normal test .
- For a single population mean , we may use a normal distribution with the following mean and standard deviation. Means: \(\mu = \mu_{\bar{x}}\) and \(\\sigma_{\bar{x}} = \frac{\sigma_{x}}{\sqrt{n}}\)
- A single population proportion , we may use a normal distribution with the following mean and standard deviation. Proportions: \(\mu = p\) and \(\sigma = \sqrt{\frac{pq}{n}}\).

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm .

- It is continuous and assumes any real values.
- The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
- It approaches the standard normal distribution as \(n\) gets larger.
- There is a "family" of \(t\)-distributions: every representative of the family is completely defined by the number of degrees of freedom which is one less than the number of data items.

## Contributors and Attributions

Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/[email protected] .

Statistics Made Easy

## Introduction to Hypothesis Testing

A statistical hypothesis is an assumption about a population parameter .

For example, we may assume that the mean height of a male in the U.S. is 70 inches.

The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter .

A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.

## The Two Types of Statistical Hypotheses

To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.

There are two types of statistical hypotheses:

The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.

The alternative hypothesis , denoted as H 1 or H a , is the hypothesis that the sample data is influenced by some non-random cause.

## Hypothesis Tests

A hypothesis test consists of five steps:

1. State the hypotheses.

State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false.

2. Determine a significance level to use for the hypothesis.

Decide on a significance level. Common choices are .01, .05, and .1.

3. Find the test statistic.

Find the test statistic and the corresponding p-value. Often we are analyzing a population mean or proportion and the general formula to find the test statistic is: (sample statistic – population parameter) / (standard deviation of statistic)

4. Reject or fail to reject the null hypothesis.

Using the test statistic or the p-value, determine if you can reject or fail to reject the null hypothesis based on the significance level.

The p-value tells us the strength of evidence in support of a null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.

5. Interpret the results.

Interpret the results of the hypothesis test in the context of the question being asked.

## The Two Types of Decision Errors

There are two types of decision errors that one can make when doing a hypothesis test:

Type I error: You reject the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level, often called alpha , and denoted as α.

Type II error: You fail to reject the null hypothesis when it is actually false. The probability of committing a Type II error is called the Power of the test or Beta , denoted as β.

## One-Tailed and Two-Tailed Tests

A statistical hypothesis can be one-tailed or two-tailed.

A one-tailed hypothesis involves making a “greater than” or “less than ” statement.

For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null hypothesis would be H0: µ ≥ 70 inches and the alternative hypothesis would be Ha: µ < 70 inches.

A two-tailed hypothesis involves making an “equal to” or “not equal to” statement.

For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches.

Note: The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.

Related: What is a Directional Hypothesis?

## Types of Hypothesis Tests

There are many different types of hypothesis tests you can perform depending on the type of data you’re working with and the goal of your analysis.

The following tutorials provide an explanation of the most common types of hypothesis tests:

Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test Introduction to the One Proportion Z-Test Introduction to the Two Proportion Z-Test

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## 9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

## Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

## Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

- H 0 : μ __ 66
- H a : μ __ 66

## Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

- H 0 : μ __ 45
- H a : μ __ 45

## Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

- H 0 : p __ 0.40
- H a : p __ 0.40

## Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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## Keyboard Shortcuts

6a.2 - steps for hypothesis tests, the logic of hypothesis testing section .

A hypothesis, in statistics, is a statement about a population parameter, where this statement typically is represented by some specific numerical value. In testing a hypothesis, we use a method where we gather data in an effort to gather evidence about the hypothesis.

How do we decide whether to reject the null hypothesis?

- If the sample data are consistent with the null hypothesis, then we do not reject it.
- If the sample data are inconsistent with the null hypothesis, but consistent with the alternative, then we reject the null hypothesis and conclude that the alternative hypothesis is true.

## Six Steps for Hypothesis Tests Section

In hypothesis testing, there are certain steps one must follow. Below these are summarized into six such steps to conducting a test of a hypothesis.

- Set up the hypotheses and check conditions : Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as \(H_0 \), which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is evidence to suggest otherwise. The second hypothesis is called the alternative, or research hypothesis, notated as \(H_a \). The alternative hypothesis is a statement of a range of alternative values in which the parameter may fall. One must also check that any conditions (assumptions) needed to run the test have been satisfied e.g. normality of data, independence, and number of success and failure outcomes.
- Decide on the significance level, \(\alpha \): This value is used as a probability cutoff for making decisions about the null hypothesis. This alpha value represents the probability we are willing to place on our test for making an incorrect decision in regards to rejecting the null hypothesis. The most common \(\alpha \) value is 0.05 or 5%. Other popular choices are 0.01 (1%) and 0.1 (10%).
- Calculate the test statistic: Gather sample data and calculate a test statistic where the sample statistic is compared to the parameter value. The test statistic is calculated under the assumption the null hypothesis is true and incorporates a measure of standard error and assumptions (conditions) related to the sampling distribution.
- Calculate probability value (p-value), or find the rejection region: A p-value is found by using the test statistic to calculate the probability of the sample data producing such a test statistic or one more extreme. The rejection region is found by using alpha to find a critical value; the rejection region is the area that is more extreme than the critical value. We discuss the p-value and rejection region in more detail in the next section.
- Make a decision about the null hypothesis: In this step, we decide to either reject the null hypothesis or decide to fail to reject the null hypothesis. Notice we do not make a decision where we will accept the null hypothesis.
- State an overall conclusion : Once we have found the p-value or rejection region, and made a statistical decision about the null hypothesis (i.e. we will reject the null or fail to reject the null), we then want to summarize our results into an overall conclusion for our test.

We will follow these six steps for the remainder of this Lesson. In the future Lessons, the steps will be followed but may not be explained explicitly.

Step 1 is a very important step to set up correctly. If your hypotheses are incorrect, your conclusion will be incorrect. In this next section, we practice with Step 1 for the one sample situations.

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Everything you need to know about the normal distribution, an in-depth explanation of cumulative distribution function, a complete guide to chi-square test, a complete guide on hypothesis testing in statistics, understanding the fundamentals of arithmetic and geometric progression, the definitive guide to understand spearman’s rank correlation, a comprehensive guide to understand mean squared error, all you need to know about the empirical rule in statistics, the complete guide to skewness and kurtosis, a holistic look at bernoulli distribution.

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## Table of Contents

In today’s data-driven world , decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

## What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life -

- A teacher assumes that 60% of his college's students come from lower-middle-class families.
- A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

## Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

- Here, x̅ is the sample mean,
- μ0 is the population mean,
- σ is the standard deviation,
- n is the sample size.

## How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

## Your Dream Career is Just Around The Corner!

## Null Hypothesis and Alternate Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average.

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

## Become a Data Scientist with Hands-on Training!

## Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine that their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

## Steps of Hypothesis Testing

Step 1: specify your null and alternate hypotheses.

It is critical to rephrase your original research hypothesis (the prediction that you wish to study) as a null (Ho) and alternative (Ha) hypothesis so that you can test it quantitatively. Your first hypothesis, which predicts a link between variables, is generally your alternate hypothesis. The null hypothesis predicts no link between the variables of interest.

## Step 2: Gather Data

For a statistical test to be legitimate, sampling and data collection must be done in a way that is meant to test your hypothesis. You cannot draw statistical conclusions about the population you are interested in if your data is not representative.

## Step 3: Conduct a Statistical Test

Other statistical tests are available, but they all compare within-group variance (how to spread out the data inside a category) against between-group variance (how different the categories are from one another). If the between-group variation is big enough that there is little or no overlap between groups, your statistical test will display a low p-value to represent this. This suggests that the disparities between these groups are unlikely to have occurred by accident. Alternatively, if there is a large within-group variance and a low between-group variance, your statistical test will show a high p-value. Any difference you find across groups is most likely attributable to chance. The variety of variables and the level of measurement of your obtained data will influence your statistical test selection.

## Step 4: Determine Rejection Of Your Null Hypothesis

Your statistical test results must determine whether your null hypothesis should be rejected or not. In most circumstances, you will base your judgment on the p-value provided by the statistical test. In most circumstances, your preset level of significance for rejecting the null hypothesis will be 0.05 - that is, when there is less than a 5% likelihood that these data would be seen if the null hypothesis were true. In other circumstances, researchers use a lower level of significance, such as 0.01 (1%). This reduces the possibility of wrongly rejecting the null hypothesis.

## Step 5: Present Your Results

The findings of hypothesis testing will be discussed in the results and discussion portions of your research paper, dissertation, or thesis. You should include a concise overview of the data and a summary of the findings of your statistical test in the results section. You can talk about whether your results confirmed your initial hypothesis or not in the conversation. Rejecting or failing to reject the null hypothesis is a formal term used in hypothesis testing. This is likely a must for your statistics assignments.

## Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

## Chi-Square

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

## Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

## Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

## One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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## Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

- The null hypothesis is (H0 <= 90) or less change.
- A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

## Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

## Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true].

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

## Level of Significance

The alpha value is a criterion for determining whether a test statistic is statistically significant. In a statistical test, Alpha represents an acceptable probability of a Type I error. Because alpha is a probability, it can be anywhere between 0 and 1. In practice, the most commonly used alpha values are 0.01, 0.05, and 0.1, which represent a 1%, 5%, and 10% chance of a Type I error, respectively (i.e. rejecting the null hypothesis when it is in fact correct).

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A p-value is a metric that expresses the likelihood that an observed difference could have occurred by chance. As the p-value decreases the statistical significance of the observed difference increases. If the p-value is too low, you reject the null hypothesis.

Here you have taken an example in which you are trying to test whether the new advertising campaign has increased the product's sales. The p-value is the likelihood that the null hypothesis, which states that there is no change in the sales due to the new advertising campaign, is true. If the p-value is .30, then there is a 30% chance that there is no increase or decrease in the product's sales. If the p-value is 0.03, then there is a 3% probability that there is no increase or decrease in the sales value due to the new advertising campaign. As you can see, the lower the p-value, the chances of the alternate hypothesis being true increases, which means that the new advertising campaign causes an increase or decrease in sales.

## Why is Hypothesis Testing Important in Research Methodology?

Hypothesis testing is crucial in research methodology for several reasons:

- Provides evidence-based conclusions: It allows researchers to make objective conclusions based on empirical data, providing evidence to support or refute their research hypotheses.
- Supports decision-making: It helps make informed decisions, such as accepting or rejecting a new treatment, implementing policy changes, or adopting new practices.
- Adds rigor and validity: It adds scientific rigor to research using statistical methods to analyze data, ensuring that conclusions are based on sound statistical evidence.
- Contributes to the advancement of knowledge: By testing hypotheses, researchers contribute to the growth of knowledge in their respective fields by confirming existing theories or discovering new patterns and relationships.

## Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

- It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
- Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
- Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
- Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore Simplilearn’s Post Graduate Program in Data Science.

If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning!

## 1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

## 2. What is hypothesis testing and its types?

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating two hypotheses: the null hypothesis (H0), which represents the default assumption, and the alternative hypothesis (Ha), which contradicts H0. The goal is to assess the evidence and determine whether there is enough statistical significance to reject the null hypothesis in favor of the alternative hypothesis.

Types of hypothesis testing:

- One-sample test: Used to compare a sample to a known value or a hypothesized value.
- Two-sample test: Compares two independent samples to assess if there is a significant difference between their means or distributions.
- Paired-sample test: Compares two related samples, such as pre-test and post-test data, to evaluate changes within the same subjects over time or under different conditions.
- Chi-square test: Used to analyze categorical data and determine if there is a significant association between variables.
- ANOVA (Analysis of Variance): Compares means across multiple groups to check if there is a significant difference between them.

## 3. What are the steps of hypothesis testing?

The steps of hypothesis testing are as follows:

- Formulate the hypotheses: State the null hypothesis (H0) and the alternative hypothesis (Ha) based on the research question.
- Set the significance level: Determine the acceptable level of error (alpha) for making a decision.
- Collect and analyze data: Gather and process the sample data.
- Compute test statistic: Calculate the appropriate statistical test to assess the evidence.
- Make a decision: Compare the test statistic with critical values or p-values and determine whether to reject H0 in favor of Ha or not.
- Draw conclusions: Interpret the results and communicate the findings in the context of the research question.

## 4. What are the 2 types of hypothesis testing?

- One-tailed (or one-sided) test: Tests for the significance of an effect in only one direction, either positive or negative.
- Two-tailed (or two-sided) test: Tests for the significance of an effect in both directions, allowing for the possibility of a positive or negative effect.

The choice between one-tailed and two-tailed tests depends on the specific research question and the directionality of the expected effect.

## 5. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

- Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
- Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
- Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

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About the author.

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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## Article Contents

- 1 Introduction
- 2 The ensemble Burden test
- 3 Extensions of the ensemble Burden test with auxiliary information about relative effect sizes
- 4 The ensemble SKAT and MORST tests
- 5 The ensemble subset chi-squared test
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- 7 Analysis of the ARIC WGS data
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## Ensemble methods for testing a global null

Conflict of interest: None declared.

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Yaowu Liu, Zhonghua Liu, Xihong Lin, Ensemble methods for testing a global null, Journal of the Royal Statistical Society Series B: Statistical Methodology , Volume 86, Issue 2, April 2024, Pages 461–486, https://doi.org/10.1093/jrsssb/qkad131

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Testing a global null is a canonical problem in statistics and has a wide range of applications. In view of the fact that no uniformly most powerful test exists, prior and/or domain knowledge are commonly used to focus on a certain class of alternatives to improve the testing power. However, it is generally challenging to develop tests that are particularly powerful against a certain class of alternatives. In this paper, motivated by the success of ensemble learning methods for prediction or classification, we propose an ensemble framework for testing that mimics the spirit of random forests to deal with the challenges. Our ensemble testing framework aggregates a collection of weak base tests to form a final ensemble test that maintains strong and robust power for global nulls. We apply the framework to four problems about global testing in different classes of alternatives arising from whole-genome sequencing (WGS) association studies. Specific ensemble tests are proposed for each of these problems, and their theoretical optimality is established in terms of Bahadur efficiency. Extensive simulations and an analysis of a real WGS dataset are conducted to demonstrate the type I error control and/or power gain of the proposed ensemble tests.

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## IMAGES

## VIDEO

## COMMENTS

An Introduction to Statistics class in Davies County, KY conducted a hypothesis test at the local high school (a medium sized-approximately 1,200 students-small city demographic) to determine if the local high school's percentage was lower. One hundred fifty students were chosen at random and surveyed.

Present the findings in your results and discussion section. Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps. Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test.

It is believed that a stock price for a particular company will grow at a rate of $5 per week with a standard deviation of $1. An investor believes the stock won't grow as quickly. The changes in stock price is recorded for ten weeks and are as follows: $4, $3, $2, $3, $1, $7, $2, $1, $1, $2. Perform a hypothesis test using a 5% level of ...

If the biologist set her significance level \(\alpha\) at 0.05 and used the critical value approach to conduct her hypothesis test, she would reject the null hypothesis if her test statistic t* were less than -1.6939 (determined using statistical software or a t-table):s-3-3. Since the biologist's test statistic, t* = -4.60, is less than -1.6939, the biologist rejects the null hypothesis.

Unit test. Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.

Example problem: A sample of 200 people has a mean age of 21 with a population standard deviation (σ) of 5. Test the hypothesis that the population mean is 18.9 at α = 0.05. Step 1: State the null hypothesis. In this case, the null hypothesis is that the population mean is 18.9, so we write: H 0: μ = 18.9. Step 2: State the alternative ...

Test Statistic: z = x¯¯¯ −μo σ/ n−−√ z = x ¯ − μ o σ / n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4 7.1. 4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.

S.3 Hypothesis Testing. In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail. The general idea of hypothesis testing involves: Making an initial assumption. Collecting evidence (data).

5.2 - Writing Hypotheses. The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ( H 0) and an alternative hypothesis ( H a ). When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the ...

Simple hypothesis testing. Niels has a Magic 8 -Ball, which is a toy used for fortune-telling or seeking advice. To consult the ball, you ask the ball a question and shake it. One of 5 different possible answers then appears at random in the ball. Niels sensed that the ball answers " Ask again later " too frequently.

Hypothesis testing is a crucial procedure to perform when you want to make inferences about a population using a random sample. These inferences include estimating population properties such as the mean, differences between means, proportions, and the relationships between variables. This post provides an overview of statistical hypothesis testing.

Statistical Inference : Hypothesis Testing: Solved Example Problems. Example 8.14. An auto company decided to introduce a new six cylinder car whose mean petrol consumption is claimed to be lower than that of the existing auto engine. It was found that the mean petrol consumption for the 50 cars was 10 km per litre with a standard deviation of ...

The above image shows a table with some of the most common test statistics and their corresponding tests or models.. A statistical hypothesis test is a method of statistical inference used to decide whether the data sufficiently support a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic.Then a decision is made, either by comparing the ...

When the research question asks "Does the independent variable affect the dependent variable?": The null hypothesis ( H0) answers "No, there's no effect in the population.". The alternative hypothesis ( Ha) answers "Yes, there is an effect in the population.". The null and alternative are always claims about the population.

In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis, typically denoted with H0.

Get the full course at: http://www.MathTutorDVD.comThe student will learn the big picture of what a hypothesis test is in statistics. We will discuss terms ...

A hypothesis test consists of five steps: 1. State the hypotheses. State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false. 2. Determine a significance level to use for the hypothesis. Decide on a significance level.

The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0, the —null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

Below these are summarized into six such steps to conducting a test of a hypothesis. Set up the hypotheses and check conditions: Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as H 0, which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is ...

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence.

This statistics video tutorial provides practice problems on hypothesis testing. It explains how to tell if you should accept or reject the null hypothesis....

Testing a global null is a canonical problem in statistics and has a wide range of applications. In view of the fact that no uniformly most powerful test exists, prior and/or domain knowledge are commonly used to focus on a certain class of alternatives to improve the testing power. However, it is generally challenging to develop tests that are ...