Inferential Statistics

# 61 Key Takeaways and Exercises

### Key Takeaways

• Null hypothesis testing is a formal approach to deciding whether a statistical relationship in a sample reflects a real relationship in the population or is just due to chance.
• The logic of null hypothesis testing involves assuming that the null hypothesis is true, finding how likely the sample result would be if this assumption were correct, and then making a decision. If the sample result would be unlikely if the null hypothesis were true, then it is rejected in favor of the alternative hypothesis. If it would not be unlikely, then the null hypothesis is retained.
• The probability of obtaining the sample result if the null hypothesis were true (the p value) is based on two considerations: relationship strength and sample size. Reasonable judgments about whether a sample relationship is statistically significant can often be made by quickly considering these two factors.
• Statistical significance is not the same as relationship strength or importance. Even weak relationships can be statistically significant if the sample size is large enough. It is important to consider relationship strength and the practical significance of a result in addition to its statistical significance.
• To compare two means, the most common null hypothesis test is the t- test. The one-sample t-test is used for comparing one sample mean with a hypothetical population mean of interest, the dependent-samples t-test is used to compare two means in a within-subjects design, and the independent-samples t-test is used to compare two means in a between-subjects design.
• To compare more than two means, the most common null hypothesis test is the analysis of variance (ANOVA). The one-way ANOVA is used for between-subjects designs with one independent variable, the repeated-measures ANOVA is used for within-subjects designs, and the factorial ANOVA is used for factorial designs.
• A null hypothesis test of Pearson’s r is used to compare a sample value of Pearson’s r with a hypothetical population value of 0.
• The decision to reject or retain the null hypothesis is not guaranteed to be correct. A Type I error occurs when one rejects the null hypothesis when it is true. A Type II error occurs when one fails to reject the null hypothesis when it is false.
• The statistical power of a research design is the probability of rejecting the null hypothesis given the expected strength of the relationship in the population and the sample size. Researchers should make sure that their studies have adequate statistical power before conducting them.
• Null hypothesis testing has been criticized on the grounds that researchers misunderstand it, that it is illogical, and that it is uninformative. Others argue that it serves an important purpose—especially when used with effect size measures, confidence intervals, and other techniques. It remains the dominant approach to inferential statistics in psychology.
• In recent years psychology has grappled with a failure to replicate research findings. Some have interpreted this as a normal aspect of science but others have suggested that this is highlights problems stemming from questionable research practices.
• One response to this “replicability crisis” has been the emergence of open science practices, which increase the transparency and openness of the research process. These open practices include digital badges to encourage pre-registration of hypotheses and the sharing of raw data and research materials.

### Exercises

• Discussion: Imagine a study showing that people who eat more broccoli tend to be happier. Explain for someone who knows nothing about statistics why the researchers would conduct a null hypothesis test.
• Practice: Use Table 13.1 to decide whether each of the following results is statistically significant.
• The correlation between two variables is r = −.78 based on a sample size of 137.
• The mean score on a psychological characteristic for women is 25 (SD = 5) and the mean score for men is 24 (SD = 5). There were 12 women and 10 men in this study.
• In a memory experiment, the mean number of items recalled by the 40 participants in Condition A was 0.50 standard deviations greater than the mean number recalled by the 40 participants in Condition B.
• In another memory experiment, the mean scores for participants in Condition A and Condition B came out exactly the same!
• A student finds a correlation of r = .04 between the number of units the students in his research methods class are taking and the students’ level of stress.
• Practice: Use one of the online tools, Excel, or SPSS to reproduce the one-sample t-test, dependent-samples t-test, independent-samples t-test, and one-way ANOVA for the four sets of calorie estimation data presented in this section.
• Practice: A sample of 25 university students rated their friendliness on a scale of 1 (Much Lower Than Average) to 7 (Much Higher Than Average). Their mean rating was 5.30 with a standard deviation of 1.50. Conduct a one-sample t-test comparing their mean rating with a hypothetical mean rating of 4 (Average). The question is whether university students have a tendency to rate themselves as friendlier than average.
• Practice: Decide whether each of the following Pearson’s r values is statistically significant for both a one-tailed and a two-tailed test.
• The correlation between height and IQ is +.13 in a sample of 35.
• For a sample of 88 university students, the correlation between how disgusted they felt and the harshness of their moral judgments was +.23.
• The correlation between the number of daily hassles and positive mood is −.43 for a sample of 30 middle-aged adults.
• Discussion: A researcher compares the effectiveness of two forms of psychotherapy for social phobia using an independent-samples t-test.
• Explain what it would mean for the researcher to commit a Type I error.
• Explain what it would mean for the researcher to commit a Type II error.
• Discussion: Imagine that you conduct a t-test and the p value is .02. How could you explain what this p value means to someone who is not already familiar with null hypothesis testing? Be sure to avoid the common misinterpretations of the p value.
• For additional practice with Type I and Type II errors, try these problems from Carnegie Mellon’s Open Learning Initiative.
• Discussion: What do you think are some of the key benefits of the adoption of open science practices such as pre-registration and the sharing of raw data and research materials? Can you identify any drawbacks of these practices?
• Practice: Read the online article “Science isn’t broken: It’s just a hell of a lot harder than we give it credit for” and use the interactive tool entitled “Hack your way to scientific glory” in order to better understand the data malpractice of “p-hacking.” 