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A study conducted at the University of Denver investigated whether babies take longer to learn to crawl in cold months, when they are often bundled in clothes that restrict their movement, than in warmer months. \({ }^{6}\) Infants born during the study year were split into twelve groups, one for each birth month. We consider the average crawling age of babies in each group against the average temperature when the babies are six months old (that's when babies often begin trying to crawl). Temperature is measured in degrees Fahrenheit ( \({ }^{\circ} \mathrm{F}\) ) and age is measured in weeks. (a) Describe the relationship between temperature and crawling age. (b) How would the relationship change if temperature was measured in degrees Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and age was measured in months? (c) The correlation between temperature in \({ }^{\circ} \mathrm{F}\) and age in weeks was \(r=-0.70\). If we converted the temperature to \({ }^{\circ} \mathrm{C}\) and age to months, what would the correlation be?

What would be the correlation between the ages of husbands and wives if men always married woman who were (a) 3 years younger than themselves? (b) 2 years older than themselves? (c) half as old as themselves?

The following regression output is for predicting the heart weight (in g) of cats from their body weight (in \(\mathrm{kg}\) ). The coefficients are estimated using a dataset of 144 domestic cats. $$\begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -0.357 & 0.692 & -0.515 & 0.607 \\ \text { body wt } & 4.034 & 0.250 & 16.119 & 0.000 \\ \hline s=1.452 & R^{2}=64.66 \% & \quad R_{a d j}^{2}=64.41 \% \end{array}$$ (a) Write out the linear model. (b) Interpret the intercept. (c) Interpret the slope. (d) Interpret \(R^{2}\). (e) Calculate the correlation coefficient.

Many people believe that gender, weight, drinking habits, and many other factors are much more important in predicting blood alcohol content (BAC) than simply considering the number of drinks a person consumed. Here we examine data from sixteen student volunteers at Ohio State University who each drank a randomly assigned number of cans of beer. These students were evenly divided between men and women, and they differed in weight and drinking habits. Thirty minutes later, a police officer measured their blood alcohol content (BAC) in grams of alcohol per deciliter of blood. \(^{19}\) The scatterplot and regression table summarize the findings. $$\begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \mathrm{t} \text { value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -0.0127 & 0.0126 & -1.00 & 0.3320 \\ \text { beers } & 0.0180 & 0.0024 & 7.48 & 0.0000 \\ \hline\end{array}$$ (a) Describe the relationship between the number of cans of beer and \(\mathrm{BAC}\). (b) Write the equation of the regression line. Interpret the slope and intercept in context. (c) Do the data provide strong evidence that drinking more cans of beer is associated with an increase in blood alcohol? State the null and alternative hypotheses, report the p-value, and state your conclusion. (d) The correlation coefficient for number of cans of beer and \(\mathrm{BAC}\) is \(0.89 .\) Calculate \(R^{2}\) and interpret it in context. (e) Suppose we visit a bar, ask people how many drinks they have had, and also take their BAC. Do you think the relationship between number of drinks and BAC would be as strong as the relationship found in the Ohio State study?

Exercise 8.26 presents regression output from a model for predicting the heart weight (in g) of cats from their body weight (in kg). The coefficients are estimated using a dataset of 144 domestic cat. The model output is also provided below. $$\begin{array}{rrrrr}\hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\\\\hline \text { (Intercept) } & -0.357 & 0.692 & -0.515 & 0.607 \\ \text { body wt } & 4.034 & 0.250 & 16.119 & 0.000 \\ \hline\end{array}$$ \(s=1.452 \quad R^{2}=64.66 \% \quad R_{a d j}^{2}=64.41 \%\) (a) We see that the point estimate for the slope is positive. What are the hypotheses for evaluating whether body weight is positively associated with heart weight in cats? (b) State the conclusion of the hypothesis test from part (a) in context of the data. (c) Calculate a \(95 \%\) confidence interval for the slope of body weight, and interpret it in context of the data. (d) Do your results from the hypothesis test and the confidence interval agree? Explain.