91Ó°ÊÓ

Two quantitative variables are described. Do you expect a positive or negative association between the two variables? Explain your choice. Amount of time spent studying and Grade on the exam

Make a scatterplot of the data. Put the \(X\) variable on the horizontal axis and the \(Y\) variable on the vertical axis. $$ \begin{array}{rrrrrrrrr} \hline X & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 \\ \hline Y & 532 & 466 & 478 & 320 & 303 & 349 & 275 & 221 \\ \hline \end{array} $$

Light Roast or Dark Roast for Your Coffee? A somewhat surprising fact about coffee is that the longer it is roasted, the less caffeine it has. Thus an "extra bold" dark roast coffee actually has less caffeine than a light roast coffee. What is the explanatory variable and what is the response variable? Do the two variables have a negative association or a positive association?

Mother's Love, Hippocampus, and Resiliency Multiple studies \(^{58}\) in both animals and humans show the importance of a mother's love (or the unconditional love of any close person to a child) in a child's brain development. A recent study shows that children with nurturing mothers had a substantially larger area of the brain called the hippocampus than children with less nurturing mothers. This is important because other studies have shown that the size of the hippocampus matters: People with large hippocampus area are more resilient and are more likely to be able to weather the stresses and strains of daily life. These observations come from experiments in animals and observational studies in humans. (a) Is the amount of maternal nurturing one receives as a child positively or negatively associated with hippocampus size? (b) Is hippocampus size positively or negatively associated with resiliency and the ability to weather the stresses of life? (c) How might a randomized experiment be designed to test the effect described in part (a) in humans? Would such an experiment be ethical? (d) Can we conclude that maternal nurturing in humans causes the hippocampus to grow larger? Can we conclude that maternal nurturing in animals (such as mice, who were used in many of the experiments) causes the hippocampus to grow larger? Explain.

Ages of Husbands and Wives Suppose we record the husband's age and the wife's age for many randomly selected couples. (a) What would it mean about ages of couples if these two variables had a negative relationship? (b) What would it mean about ages of couples if these two variables had a positive relationship? (c) Which do you think is more likely, a negative or a positive relationship? (d) Do you expect a strong or a weak relationship in the data? Why? (e) Would a strong correlation imply there is an association between husband age and wife age?

Is Your Body Language Closed or Open? A closed body posture includes sitting hunched over or standing with arms crossed rather than sitting or standing up straight and having the arms more open. According to a recent study, people who were rated as having a more closed body posture "had higher levels of stress hormones and said they felt less powerful than those who had a more open pose." (a) What are the variables in this study? Is each variable categorical or quantitative? Assume participants had body language rated on a numerical scale from low values representing more closed to larger values representing more open. Assume also that participants were rated on a numerical scale indicating whether each felt less powerful (low values) or more powerful (higher values). (b) Do the results of the study indicate a positive or negative relationship between the body language scores and levels of stress hormones? Would your answer be different if the scale had been reversed for the body language scores? (c) Do the results of the study indicate a positive or negative relationship between the body language scores and the scores on the feelings of power? Would your answer be different if both scales were reversed? Would your answer be different if only one of the scales had been reversed?

Do Movies with Larger Budgets Get Higher Audience Ratings? The dataset HollywoodMovies2011 is introduced on page \(93,\) and includes many variables for movies that were produced in Hollywood in 2011, including Budget and AudienceScore. (a) Use technology to create a scatterplot to show the relationship between the budget of a movie, in millions of dollars, and the audience score. We want to see if the budget has an effect on the audience score. (b) Is there a linear relationship? How strong is it? Give your answer in the context of movies. (c) There is an outlier with a very large budget. What is the audience rating for this movie and what movie is it? There is another data value with a budget of about 125 million dollars and an audience score over 90 . To what movie does that dot correspond? (d) Use technology to find the correlation between these two variables.

Two variables are defined, a regression equation is given, and one data point is given. (a) Find the predicted value for the data point and compute the residual. (b) Interpret the slope in context. (c) Interpret the intercept in context, and if the intercept makes no sense in this context, explain why. Weight \(=\) maximum weight capable of bench pressing (pounds), Training \(=\) number of hours spent lifting weights a week \(\widehat{\text { Weigh }} t=95+11.7\) (Training); data point is an individual who trains 5 hours a week and can bench 150 pounds

Height and Weight Using the data in the StudentSurvey dataset, we use technology to find that a regression line to predict weight (in pounds) from height (in inches) is $$ \widehat{\text { Weigh }} t=-170+4.82(\text { Height }) $$ (a) What weight does the line predict for a person who is 5 feet tall ( 60 inches)? What weight is predicted for someone 6 feet tall ( 72 inches)? (b) What is the slope of the line? Interpret it in context. (c) What is the intercept of the line? If it is reasonable to do so, interpret it in context. If it is not reasonable, explain why not. (d) What weight does the regression line predict for a baby who is 20 inches long? Why is it not appropriate to use the regression line in this case?

Pick a Relationship to Examine Choose one of the following datasets: USStates, StudentSurvey, AllCountries, or NBAPlayers2011, and then select any two quantitative variables that we have not yet analyzed. Use technology to create a scatterplot of the two variables with the regression line on it and discuss what you see. If there is a reasonable linear relationship, find a formula for the regression line. If not, find two other quantitative variables that do have a reasonable linear relationship and find the regression line for them. Indicate whether there are any outliers in the dataset that might be influential points or have large residuals. Be sure to state the dataset and variables you use.