10.31 Temperature and academic performance. Does temperature affect academic performance? If yes, does the relationship vary by sex? To study these questions, researchers from Berlin, Germany, divided 543 students into 24 sessions. In each session, students were presented with 50 similar arithmetic problems and given 5 minutes to complete as many as possible. (They were monetarily rewarded for the number of correct answers.) The sessions varied only in terms of the room temperature, which ranged from 16.19 to 32.57°C. Although the number of male and female students varied across sessions, let’s look at the relationship between each session’s room temperature (Temp) and the average number of correct answers by sex (Mave and Fave).

a. Make a scatterplot of Mave versus Temp. Describe the relationship.

b. Find the equation of the least-squares regression line for predicting Mave based on the room temperature and add this line to your scatterplot.

c. What is r2 for these data? Briefly explain what this tells you about the overall fit of the model to these data.

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d. Check the conditions that must be approximately met for inference. Provide a set of plots and any concerns you have.

e. Assuming that inference is appropriate, is there significant evidence that temperature is associated with performance? State the hypotheses, give a test statistic and P-value, and summarize your conclusion.