conclusion of statistics assignment

Psychological Research

Drawing Conclusions from Statistics

Learning objectives.

• Describe the role of random sampling and random assignment in drawing cause-and-effect conclusions

Generalizability

One limitation to the study mentioned previously about the babies choosing the “helper” toy is that the conclusion only applies to the 16 infants in the study. We don’t know much about how those 16 infants were selected. Suppose we want to select a subset of individuals (a sample ) from a much larger group of individuals (the population ) in such a way that conclusions from the sample can be generalized to the larger population. This is the question faced by pollsters every day.

Example 1 : The General Social Survey (GSS) is a survey on societal trends conducted every other year in the United States. Based on a sample of about 2,000 adult Americans, researchers make claims about what percentage of the U.S. population consider themselves to be “liberal,” what percentage consider themselves “happy,” what percentage feel “rushed” in their daily lives, and many other issues. The key to making these claims about the larger population of all American adults lies in how the sample is selected. The goal is to select a sample that is representative of the population, and a common way to achieve this goal is to select a random sample that gives every member of the population an equal chance of being selected for the sample. In its simplest form, random sampling involves numbering every member of the population and then using a computer to randomly select the subset to be surveyed. Most polls don’t operate exactly like this, but they do use probability-based sampling methods to select individuals from nationally representative panels.

In 2004, the GSS reported that 817 of 977 respondents (or 83.6%) indicated that they always or sometimes feel rushed. This is a clear majority, but we again need to consider variation due to random sampling . Fortunately, we can use the same probability model we did in the previous example to investigate the probable size of this error. (Note, we can use the coin-tossing model when the actual population size is much, much larger than the sample size, as then we can still consider the probability to be the same for every individual in the sample.) This probability model predicts that the sample result will be within 3 percentage points of the population value (roughly 1 over the square root of the sample size, the margin of error ). A statistician would conclude, with 95% confidence, that between 80.6% and 86.6% of all adult Americans in 2004 would have responded that they sometimes or always feel rushed.

The key to the margin of error is that when we use a probability sampling method, we can make claims about how often (in the long run, with repeated random sampling) the sample result would fall within a certain distance from the unknown population value by chance (meaning by random sampling variation) alone. Conversely, non-random samples are often suspect to bias, meaning the sampling method systematically over-represents some segments of the population and under-represents others. We also still need to consider other sources of bias, such as individuals not responding honestly. These sources of error are not measured by the margin of error.

Cause and Effect

In many research studies, the primary question of interest concerns differences between groups. Then the question becomes how were the groups formed (e.g., selecting people who already drink coffee vs. those who don’t). In some studies, the researchers actively form the groups themselves. But then we have a similar question—could any differences we observe in the groups be an artifact of that group-formation process? Or maybe the difference we observe in the groups is so large that we can discount a “fluke” in the group-formation process as a reasonable explanation for what we find?

Example 2 : A psychology study investigated whether people tend to display more creativity when they are thinking about intrinsic (internal) or extrinsic (external) motivations (Ramsey & Schafer, 2002, based on a study by Amabile, 1985). The subjects were 47 people with extensive experience with creative writing. Subjects began by answering survey questions about either intrinsic motivations for writing (such as the pleasure of self-expression) or extrinsic motivations (such as public recognition). Then all subjects were instructed to write a haiku, and those poems were evaluated for creativity by a panel of judges. The researchers conjectured beforehand that subjects who were thinking about intrinsic motivations would display more creativity than subjects who were thinking about extrinsic motivations. The creativity scores from the 47 subjects in this study are displayed in Figure 2, where higher scores indicate more creativity.

In this example, the key question is whether the type of motivation affects creativity scores. In particular, do subjects who were asked about intrinsic motivations tend to have higher creativity scores than subjects who were asked about extrinsic motivations?

Figure 2 reveals that both motivation groups saw considerable variability in creativity scores, and these scores have considerable overlap between the groups. In other words, it’s certainly not always the case that those with extrinsic motivations have higher creativity than those with intrinsic motivations, but there may still be a statistical tendency in this direction. (Psychologist Keith Stanovich (2013) refers to people’s difficulties with thinking about such probabilistic tendencies as “the Achilles heel of human cognition.”)

The mean creativity score is 19.88 for the intrinsic group, compared to 15.74 for the extrinsic group, which supports the researchers’ conjecture. Yet comparing only the means of the two groups fails to consider the variability of creativity scores in the groups. We can measure variability with statistics using, for instance, the standard deviation: 5.25 for the extrinsic group and 4.40 for the intrinsic group. The standard deviations tell us that most of the creativity scores are within about 5 points of the mean score in each group. We see that the mean score for the intrinsic group lies within one standard deviation of the mean score for extrinsic group. So, although there is a tendency for the creativity scores to be higher in the intrinsic group, on average, the difference is not extremely large.

We again want to consider possible explanations for this difference. The study only involved individuals with extensive creative writing experience. Although this limits the population to which we can generalize, it does not explain why the mean creativity score was a bit larger for the intrinsic group than for the extrinsic group. Maybe women tend to receive higher creativity scores? Here is where we need to focus on how the individuals were assigned to the motivation groups. If only women were in the intrinsic motivation group and only men in the extrinsic group, then this would present a problem because we wouldn’t know if the intrinsic group did better because of the different type of motivation or because they were women. However, the researchers guarded against such a problem by randomly assigning the individuals to the motivation groups. Like flipping a coin, each individual was just as likely to be assigned to either type of motivation. Why is this helpful? Because this random assignment tends to balance out all the variables related to creativity we can think of, and even those we don’t think of in advance, between the two groups. So we should have a similar male/female split between the two groups; we should have a similar age distribution between the two groups; we should have a similar distribution of educational background between the two groups; and so on. Random assignment should produce groups that are as similar as possible except for the type of motivation, which presumably eliminates all those other variables as possible explanations for the observed tendency for higher scores in the intrinsic group.

But does this always work? No, so by “luck of the draw” the groups may be a little different prior to answering the motivation survey. So then the question is, is it possible that an unlucky random assignment is responsible for the observed difference in creativity scores between the groups? In other words, suppose each individual’s poem was going to get the same creativity score no matter which group they were assigned to, that the type of motivation in no way impacted their score. Then how often would the random-assignment process alone lead to a difference in mean creativity scores as large (or larger) than 19.88 – 15.74 = 4.14 points?

We again want to apply to a probability model to approximate a p-value , but this time the model will be a bit different. Think of writing everyone’s creativity scores on an index card, shuffling up the index cards, and then dealing out 23 to the extrinsic motivation group and 24 to the intrinsic motivation group, and finding the difference in the group means. We (better yet, the computer) can repeat this process over and over to see how often, when the scores don’t change, random assignment leads to a difference in means at least as large as 4.41. Figure 3 shows the results from 1,000 such hypothetical random assignments for these scores.

Only 2 of the 1,000 simulated random assignments produced a difference in group means of 4.41 or larger. In other words, the approximate p-value is 2/1000 = 0.002. This small p-value indicates that it would be very surprising for the random assignment process alone to produce such a large difference in group means. Therefore, as with Example 2, we have strong evidence that focusing on intrinsic motivations tends to increase creativity scores, as compared to thinking about extrinsic motivations.

Notice that the previous statement implies a cause-and-effect relationship between motivation and creativity score; is such a strong conclusion justified? Yes, because of the random assignment used in the study. That should have balanced out any other variables between the two groups, so now that the small p-value convinces us that the higher mean in the intrinsic group wasn’t just a coincidence, the only reasonable explanation left is the difference in the type of motivation. Can we generalize this conclusion to everyone? Not necessarily—we could cautiously generalize this conclusion to individuals with extensive experience in creative writing similar the individuals in this study, but we would still want to know more about how these individuals were selected to participate.

Statistical thinking involves the careful design of a study to collect meaningful data to answer a focused research question, detailed analysis of patterns in the data, and drawing conclusions that go beyond the observed data. Random sampling is paramount to generalizing results from our sample to a larger population, and random assignment is key to drawing cause-and-effect conclusions. With both kinds of randomness, probability models help us assess how much random variation we can expect in our results, in order to determine whether our results could happen by chance alone and to estimate a margin of error.

So where does this leave us with regard to the coffee study mentioned previously (the Freedman, Park, Abnet, Hollenbeck, & Sinha, 2012 found that men who drank at least six cups of coffee a day had a 10% lower chance of dying (women 15% lower) than those who drank none)? We can answer many of the questions:

• This was a 14-year study conducted by researchers at the National Cancer Institute.
• The results were published in the June issue of the New England Journal of Medicine , a respected, peer-reviewed journal.
• The study reviewed coffee habits of more than 402,000 people ages 50 to 71 from six states and two metropolitan areas. Those with cancer, heart disease, and stroke were excluded at the start of the study. Coffee consumption was assessed once at the start of the study.
• About 52,000 people died during the course of the study.
• People who drank between two and five cups of coffee daily showed a lower risk as well, but the amount of reduction increased for those drinking six or more cups.
• The sample sizes were fairly large and so the p-values are quite small, even though percent reduction in risk was not extremely large (dropping from a 12% chance to about 10%–11%).
• Whether coffee was caffeinated or decaffeinated did not appear to affect the results.
• This was an observational study, so no cause-and-effect conclusions can be drawn between coffee drinking and increased longevity, contrary to the impression conveyed by many news headlines about this study. In particular, it’s possible that those with chronic diseases don’t tend to drink coffee.

This study needs to be reviewed in the larger context of similar studies and consistency of results across studies, with the constant caution that this was not a randomized experiment. Whereas a statistical analysis can still “adjust” for other potential confounding variables, we are not yet convinced that researchers have identified them all or completely isolated why this decrease in death risk is evident. Researchers can now take the findings of this study and develop more focused studies that address new questions.

Explore these outside resources to learn more about applied statistics:

• Video about p-values:  P-Value Extravaganza
• Interactive web applets for teaching and learning statistics
• Inter-university Consortium for Political and Social Research  where you can find and analyze data.
• The Consortium for the Advancement of Undergraduate Statistics

Think It Over

• Find a recent research article in your field and answer the following: What was the primary research question? How were individuals selected to participate in the study? Were summary results provided? How strong is the evidence presented in favor or against the research question? Was random assignment used? Summarize the main conclusions from the study, addressing the issues of statistical significance, statistical confidence, generalizability, and cause and effect. Do you agree with the conclusions drawn from this study, based on the study design and the results presented?
• Is it reasonable to use a random sample of 1,000 individuals to draw conclusions about all U.S. adults? Explain why or why not.

CC licensed content, Original

• Modification, adaptation, and original content. Authored by : Pat Carroll and Lumen Learning. Provided by : Lumen Learning. License : CC BY: Attribution

CC licensed content, Shared previously

• Statistical Thinking. Authored by : Beth Chance and Allan Rossman, California Polytechnic State University, San Luis Obispo. Provided by : Noba. Located at : http://nobaproject.com/modules/statistical-thinking . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
• The Replication Crisis. Authored by : Colin Thomas William. Provided by : Ivy Tech Community College. License : CC BY: Attribution

related to whether the results from the sample can be generalized to a larger population.

the collection of individuals on which we collect data.

a larger collection of individuals that we would like to generalize our results to.

using a probability-based method to select a subset of individuals for the sample from the population.

the expected amount of random variation in a statistic; often defined for 95% confidence level.

using a probability-based method to divide a sample into treatment groups.

the probability of observing a particular outcome in a sample, or more extreme, under a conjecture about the larger population or process.

related to whether we say one variable is causing changes in the other variable, versus other variables that may be related to these two variables.

General Psychology Copyright © by OpenStax and Lumen Learning is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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5.15: Drawing Conclusions from Statistics

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Learning Objectives

• Describe the role of random sampling and random assignment in drawing cause-and-effect conclusions

Generalizability

One limitation to the study mentioned previously about the babies choosing the “helper” toy is that the conclusion only applies to the 16 infants in the study. We don’t know much about how those 16 infants were selected. Suppose we want to select a subset of individuals (a sample ) from a much larger group of individuals (the population ) in such a way that conclusions from the sample can be generalized to the larger population. This is the question faced by pollsters every day.

Example 1 : The General Social Survey (GSS) is a survey on societal trends conducted every other year in the United States. Based on a sample of about 2,000 adult Americans, researchers make claims about what percentage of the U.S. population consider themselves to be “liberal,” what percentage consider themselves “happy,” what percentage feel “rushed” in their daily lives, and many other issues. The key to making these claims about the larger population of all American adults lies in how the sample is selected. The goal is to select a sample that is representative of the population, and a common way to achieve this goal is to select a random sample that gives every member of the population an equal chance of being selected for the sample. In its simplest form, random sampling involves numbering every member of the population and then using a computer to randomly select the subset to be surveyed. Most polls don’t operate exactly like this, but they do use probability-based sampling methods to select individuals from nationally representative panels.

In 2004, the GSS reported that 817 of 977 respondents (or 83.6%) indicated that they always or sometimes feel rushed. This is a clear majority, but we again need to consider variation due to random sampling . Fortunately, we can use the same probability model we did in the previous example to investigate the probable size of this error. (Note, we can use the coin-tossing model when the actual population size is much, much larger than the sample size, as then we can still consider the probability to be the same for every individual in the sample.) This probability model predicts that the sample result will be within 3 percentage points of the population value (roughly 1 over the square root of the sample size, the margin of error ). A statistician would conclude, with 95% confidence, that between 80.6% and 86.6% of all adult Americans in 2004 would have responded that they sometimes or always feel rushed.

The key to the margin of error is that when we use a probability sampling method, we can make claims about how often (in the long run, with repeated random sampling) the sample result would fall within a certain distance from the unknown population value by chance (meaning by random sampling variation) alone. Conversely, non-random samples are often suspect to bias, meaning the sampling method systematically over-represents some segments of the population and under-represents others. We also still need to consider other sources of bias, such as individuals not responding honestly. These sources of error are not measured by the margin of error.

Query \(\PageIndex{1}\)

Query \(\PageIndex{2}\)

Cause and Effect

In many research studies, the primary question of interest concerns differences between groups. Then the question becomes how were the groups formed (e.g., selecting people who already drink coffee vs. those who don’t). In some studies, the researchers actively form the groups themselves. But then we have a similar question—could any differences we observe in the groups be an artifact of that group-formation process? Or maybe the difference we observe in the groups is so large that we can discount a “fluke” in the group-formation process as a reasonable explanation for what we find?

Example 2 : A psychology study investigated whether people tend to display more creativity when they are thinking about intrinsic (internal) or extrinsic (external) motivations (Ramsey & Schafer, 2002, based on a study by Amabile, 1985). The subjects were 47 people with extensive experience with creative writing. Subjects began by answering survey questions about either intrinsic motivations for writing (such as the pleasure of self-expression) or extrinsic motivations (such as public recognition). Then all subjects were instructed to write a haiku, and those poems were evaluated for creativity by a panel of judges. The researchers conjectured beforehand that subjects who were thinking about intrinsic motivations would display more creativity than subjects who were thinking about extrinsic motivations. The creativity scores from the 47 subjects in this study are displayed in Figure 2, where higher scores indicate more creativity.

In this example, the key question is whether the type of motivation affects creativity scores. In particular, do subjects who were asked about intrinsic motivations tend to have higher creativity scores than subjects who were asked about extrinsic motivations?

Figure 2 reveals that both motivation groups saw considerable variability in creativity scores, and these scores have considerable overlap between the groups. In other words, it’s certainly not always the case that those with extrinsic motivations have higher creativity than those with intrinsic motivations, but there may still be a statistical tendency in this direction. (Psychologist Keith Stanovich (2013) refers to people’s difficulties with thinking about such probabilistic tendencies as “the Achilles heel of human cognition.”)

The mean creativity score is 19.88 for the intrinsic group, compared to 15.74 for the extrinsic group, which supports the researchers’ conjecture. Yet comparing only the means of the two groups fails to consider the variability of creativity scores in the groups. We can measure variability with statistics using, for instance, the standard deviation: 5.25 for the extrinsic group and 4.40 for the intrinsic group. The standard deviations tell us that most of the creativity scores are within about 5 points of the mean score in each group. We see that the mean score for the intrinsic group lies within one standard deviation of the mean score for extrinsic group. So, although there is a tendency for the creativity scores to be higher in the intrinsic group, on average, the difference is not extremely large.

We again want to consider possible explanations for this difference. The study only involved individuals with extensive creative writing experience. Although this limits the population to which we can generalize, it does not explain why the mean creativity score was a bit larger for the intrinsic group than for the extrinsic group. Maybe women tend to receive higher creativity scores? Here is where we need to focus on how the individuals were assigned to the motivation groups. If only women were in the intrinsic motivation group and only men in the extrinsic group, then this would present a problem because we wouldn’t know if the intrinsic group did better because of the different type of motivation or because they were women. However, the researchers guarded against such a problem by randomly assigning the individuals to the motivation groups. Like flipping a coin, each individual was just as likely to be assigned to either type of motivation. Why is this helpful? Because this random assignment tends to balance out all the variables related to creativity we can think of, and even those we don’t think of in advance, between the two groups. So we should have a similar male/female split between the two groups; we should have a similar age distribution between the two groups; we should have a similar distribution of educational background between the two groups; and so on. Random assignment should produce groups that are as similar as possible except for the type of motivation, which presumably eliminates all those other variables as possible explanations for the observed tendency for higher scores in the intrinsic group.

But does this always work? No, so by “luck of the draw” the groups may be a little different prior to answering the motivation survey. So then the question is, is it possible that an unlucky random assignment is responsible for the observed difference in creativity scores between the groups? In other words, suppose each individual’s poem was going to get the same creativity score no matter which group they were assigned to, that the type of motivation in no way impacted their score. Then how often would the random-assignment process alone lead to a difference in mean creativity scores as large (or larger) than 19.88 – 15.74 = 4.14 points?

We again want to apply to a probability model to approximate a p-value , but this time the model will be a bit different. Think of writing everyone’s creativity scores on an index card, shuffling up the index cards, and then dealing out 23 to the extrinsic motivation group and 24 to the intrinsic motivation group, and finding the difference in the group means. We (better yet, the computer) can repeat this process over and over to see how often, when the scores don’t change, random assignment leads to a difference in means at least as large as 4.41. Figure 3 shows the results from 1,000 such hypothetical random assignments for these scores.

Only 2 of the 1,000 simulated random assignments produced a difference in group means of 4.41 or larger. In other words, the approximate p-value is 2/1000 = 0.002. This small p-value indicates that it would be very surprising for the random assignment process alone to produce such a large difference in group means. Therefore, as with Example 2, we have strong evidence that focusing on intrinsic motivations tends to increase creativity scores, as compared to thinking about extrinsic motivations.

Notice that the previous statement implies a cause-and-effect relationship between motivation and creativity score; is such a strong conclusion justified? Yes, because of the random assignment used in the study. That should have balanced out any other variables between the two groups, so now that the small p-value convinces us that the higher mean in the intrinsic group wasn’t just a coincidence, the only reasonable explanation left is the difference in the type of motivation. Can we generalize this conclusion to everyone? Not necessarily—we could cautiously generalize this conclusion to individuals with extensive experience in creative writing similar the individuals in this study, but we would still want to know more about how these individuals were selected to participate.

Statistical thinking involves the careful design of a study to collect meaningful data to answer a focused research question, detailed analysis of patterns in the data, and drawing conclusions that go beyond the observed data. Random sampling is paramount to generalizing results from our sample to a larger population, and random assignment is key to drawing cause-and-effect conclusions. With both kinds of randomness, probability models help us assess how much random variation we can expect in our results, in order to determine whether our results could happen by chance alone and to estimate a margin of error.

So where does this leave us with regard to the coffee study mentioned previously (the Freedman, Park, Abnet, Hollenbeck, & Sinha, 2012 found that men who drank at least six cups of coffee a day had a 10% lower chance of dying (women 15% lower) than those who drank none)? We can answer many of the questions:

• This was a 14-year study conducted by researchers at the National Cancer Institute.
• The results were published in the June issue of the New England Journal of Medicine , a respected, peer-reviewed journal.
• The study reviewed coffee habits of more than 402,000 people ages 50 to 71 from six states and two metropolitan areas. Those with cancer, heart disease, and stroke were excluded at the start of the study. Coffee consumption was assessed once at the start of the study.
• About 52,000 people died during the course of the study.
• People who drank between two and five cups of coffee daily showed a lower risk as well, but the amount of reduction increased for those drinking six or more cups.
• The sample sizes were fairly large and so the p-values are quite small, even though percent reduction in risk was not extremely large (dropping from a 12% chance to about 10%–11%).
• Whether coffee was caffeinated or decaffeinated did not appear to affect the results.
• This was an observational study, so no cause-and-effect conclusions can be drawn between coffee drinking and increased longevity, contrary to the impression conveyed by many news headlines about this study. In particular, it’s possible that those with chronic diseases don’t tend to drink coffee.

This study needs to be reviewed in the larger context of similar studies and consistency of results across studies, with the constant caution that this was not a randomized experiment. Whereas a statistical analysis can still “adjust” for other potential confounding variables, we are not yet convinced that researchers have identified them all or completely isolated why this decrease in death risk is evident. Researchers can now take the findings of this study and develop more focused studies that address new questions.

Explore these outside resources to learn more about applied statistics:

• Video about p-values:  P-Value Extravaganza
• Interactive web applets for teaching and learning statistics
• Inter-university Consortium for Political and Social Research  where you can find and analyze data.
• The Consortium for the Advancement of Undergraduate Statistics

Think It Over

• Find a recent research article in your field and answer the following: What was the primary research question? How were individuals selected to participate in the study? Were summary results provided? How strong is the evidence presented in favor or against the research question? Was random assignment used? Summarize the main conclusions from the study, addressing the issues of statistical significance, statistical confidence, generalizability, and cause and effect. Do you agree with the conclusions drawn from this study, based on the study design and the results presented?
• Is it reasonable to use a random sample of 1,000 individuals to draw conclusions about all U.S. adults? Explain why or why not.

cause-and-effect: related to whether we say one variable is causing changes in the other variable, versus other variables that may be related to these two variables.

generalizability : related to whether the results from the sample can be generalized to a larger population.

margin of error : the expected amount of random variation in a statistic; often defined for 95% confidence level.

population : a larger collection of individuals that we would like to generalize our results to.

p-value : the probability of observing a particular outcome in a sample, or more extreme, under a conjecture about the larger population or process.

random assignment : using a probability-based method to divide a sample into treatment groups.

random sampling : using a probability-based method to select a subset of individuals for the sample from the population.

sample : the collection of individuals on which we collect data.

Licenses and Attributions

CC licensed content, Original

• Modification, adaptation, and original content. Authored by : Pat Carroll and Lumen Learning. Provided by : Lumen Learning. License : CC BY: Attribution
• Statistical Thinking. Authored by : Beth Chance and Allan Rossman, California Polytechnic State University, San Luis Obispo. Provided by : Noba. Located at : http://nobaproject.com/modules/statistical-thinking . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
• The Replication Crisis. Authored by : Colin Thomas William. Provided by : Ivy Tech Community College. License : CC BY: Attribution

Statistics Resources

• Excel - Tutorials
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• Quantitative Research Questions
• Null & Alternative Hypotheses
• One-Tail vs. Two-Tail
• Alpha & Beta
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• Decision Rule
• Statement of Conclusion
• Statistics Group Sessions

Statement of the Conclusion

When writing your results, you’re going to write the decision regarding the null, but you also want to state the results in layman’s terms. Tie the statistical results back to the original claim and interpret what those statistics mean, without all the quantitative jargon.

1)  Claim : Females run faster than males.             Results of the test : t o > t c             Decision : Reject Null Hypothesis.             Conclusion : There is sufficient evidence to suggest that females run faster than males.

2)  Claim : There is a difference in the highest level of education obtained based on socioeconomic status. Results of the test : p > α Decision : Fail to Reject Null Hypothesis. Conclusion : There is not enough evidence to suggest that highest level of education differs based on socioeconomic status.

3)  Claim : The number of calories consumed and the number of hours spent exercising each week are significant predictors of weight. Results of the test : p < α Decision : Reject Null Hypothesis. Conclusion : The results of the hypothesis test suggest that a person’s weight can be predicted given caloric intake and the number of hours spent exercising each week.

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Statistical Thinking: A Simulation Approach to Modeling Uncertainty (UM STAT 216 edition)

2.14 drawing conclusions and “statistical significance”.

We have seen that statistical hypothesis testing is a process of comparing the real-world observed result to a null hypothesis where there is no effect . At the end of the process, we compare the observed result to the distribution of simulated results if the null hypothesis were true, and from this we determine whether the observed result is compatible with the null hypothesis.

The conclusions that we can draw form a hypothesis test are based on the comparison between the observed result and the null hypothesis. For example, in the Monday breakups study , we concluded:

The observed result is not compatible with the null hypothesis. This suggests that breakups may be more likely to be reported on Monday.

There are two important point to notice in how this conclusion is written:

• The conclusion is stated in terms of compatibility with the null hypothesis .
• The conclusion uses soft language like “suggests.” This is becuase we did not prove that breakups are more likely to be reported on Monday. Instead, we simply have strong evidence against the null hypothesis (that breakups are equally likely each day). This, in turn, suggests that breakups are more likely to be reported on Mondays.

Similarly, if the observed result had been within the range of likely results if the null hypothesis were true, we would still write the conclusion in terms of compatibility with the null hypothesis:

The observed result is compatible with the null hypothesis. We do not have sufficient evidence to suggest that breakups are more likely to be reported on Monday.

In both cases, notice that the conclusion is limited to whether there is an effect or not. There are many additional aspects that we might be interested in, but the hypothesis test does not tell us about. For example,

• We don’t know what caused the effect.
• We don’t know the size of the effect. Perhaps the true percentage of Monday breakups is 26%. Perhaps it is slightly more or slightly less. We only have evidence that the results are incompatible with the null hypothesis.
• We don’t know the scope of the effect. Perhaps the phenomenon is limited to this particular year, or to breakups that are reported on facebook, etc.

(We will learn about size, scope, and causation later in the course. The key point to understand now is that a hypothesis test, by itself, can not tell us about these things and so the conclusion should not address them.)

2.14.1 Statistical significance

In news reports and scientific literature, we often hear the term, “statistical significance.” What does it mean for a result to be “statistically significant?” In short, it means that the observed result is not compatible with the null hypothesis.

Different scientific communities have different standards for determining whether a result is statistically significant. In the social sciences, there are two common approaches for determining statistical significance.

• Use the range of likely results: The first approach is to determine whether the observed result is within the range of likely results if the null hypothesis were true. If the observed result is outside the range of likely values if the null hypothesis were true, then social scientists consider A second common practice is to use \(p\) -values. Commonly, social scientists consider that to be sufficient evidence that the observed result is not compatible with the null hypothesis, and thus that the observed result is statistically significant.
• Use p < 0.05: A second common approach is to use a \(p\) -value of 0.05 as a threshold. If \(p<0.05\) , social scientists consider that to be sufficient evidence that the observed result is not compatible with the null hypothesis, and thus that the observed result is statistically significant.

Other scientific communities may have different standards. Moreover, there is currently a lot of discussion about whether the current thresholds should be reconsidered, and even whether we should even have a threshold. Some scholars advocate that researchers should just report the \(p\) -value and make an argument as to whether it provides sufficient evidence against the null model.

For our class, you can use either the “range of likely values” approach, the “ \(p<0.05\) ” approach, or the “report the p-value and make an argument” approach to determining whether an observed result is statistically significant. As you become a member of a scientific community, you will learn which approaches that community uses.

2.14.2 Statistical significance vs. practical significance

Don’t confuse statistical significance with practical significance. Often, statistical significance is taken to be a indication of whether the result is meaningful in the real world (i.e., “practically significant”). But statistical significance has nothing to do with real-world importance. Remember, statistical significance just tells us whether the observed result is compatible with the null hypothesis. The question of whether the result is of real-world (or practical) significance cannot be determined statistically. Instead, this is something that people have to make an argument about.

2.14.3 Other things that statistical significance can’t tell us.

Again, statistical significance only tells us that an observed result is not compatible with the null hypothesis. It does not tell us about other important aspects, including:

• Statistical significance does not mean that we have proven something. It only tells us that the there is evidence against a null model, which in turn would suggest that the effect is real.
• Statistical significance says nothing about what caused the effect
• Statistical significance does not tell us the scope of the effect (that is, how broadly the result apply).

2.14.4 Examples

Here is how to write a conclusion to a hypothesis test.

If the result is statistically significant:

The observed result is not compatible with the null hypothesis. This suggests that there may be an effect.

If the result is not statistically significant:

The observed result is compatible with the null hypothesis. We do not have sufficient evidence to suggest that there is an effect.

2.14.5 Summary

The box below summarizes the key points about drawing conclusions and statistical significance. statistical hypothesis testing.

Key points about drawing conclusions and statistical significance

Conclusions from a hypothesis test are stated in terms of compatibility with the null hypothesis

We do not prove anything, so conclusions should use softer language like suggests

Statistical significance simply means that the observed result is not compatible with the null hypothesis

• Statistical significance does not tell us the size of the effect, or whether it is large enough to have real-world importance.

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1: Introduction to Statistics

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In this chapter we will introduce some basic terminology and lay the groundwork for the course. We will explain in general terms what statistics and probability are and the problems that these two areas of study are designed to solve.

• 1.1: Basic Definitions and Concepts Statistics is a study of data: describing properties of data (descriptive statistics) and drawing conclusions about a population based on information in a sample (inferential statistics). The distinction between a population together with its parameters and a sample together with its statistics is a fundamental concept in inferential statistics. Information in a sample is used to make inferences about the population from which the sample was drawn.
• 1.2: Overview Statistics computed from samples vary randomly from sample to sample. Conclusions made about population parameters are statements of probability.
• 1.3: Presentation of Data In this book we will use two formats for presenting data sets. Data could be presented as the data list or in set notation.
• 1.E: Introduction to Statistics (Exercises) These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang.