For a sample of $41 \mathrm{New}$ England cities, a sociologist studies the crime rate in each city (crimes per 100,000 residents) as a function of its poverty rate (in %) and its median income (in $\$ 1,000$ s). He finds that $S S E=4,182,663$ and $S S T=7,732,451$.

b. What proportion of the sample variation in crime rate is explained by the variability in the explanatory variables? What proportion is unexplained?

A sample of 10 observations provides the following statistics:

$$s_x=13, \quad s_y=18, \quad \text { and } s_{x y}=117.22$$

a. Calculate and interpret the sample correlation coefficient $r_{x y}$.

b. Specify the competing hypotheses to determine whether the population correlation coefficient is positive.

c. Calculate the value of the test statistic and the $p$-value.

d. At the $5 \%$ significance level, what is the conclusion to the test? Explain.

Healthy living has always been an important goal for any society. In a recent ad campaign for Walt Disney, First Lady Michelle Obama shows parents and children that eating well and exercising can also be fun (USA Today, September 30, 2010). Consider a regression model that conjectures that fruits and vegetables and regular exercising have a positive effect on health and smoking has a negative effect on health. The sample consists of the percentage of these variables observed in various states in the United States in 2009. A portion of the data is shown in the accompanying table.

$$\begin{array}{|c|c|c|c|c|} \hline \text { State } & \begin{array}{c} \text { Healthy } \\ (\%) \end{array} & \begin{array}{c} \text { Fruits/Vegetables } \\ (\%) \end{array} & \begin{array}{c} \text { Exercise } \\ (\%) \end{array} & \begin{array}{c} \text { Smoke } \\ (\%) \end{array} \\ \hline \text { AK } & 88.7 & 23.3 & 60.6 & 14.6 \\ \hline \text { AL } & 78.3 & 20.3 & 41 & 16.4 \\ \hline \vdots & \vdots & \vdots & \vdots & \vdots \\ \hline \text { WY } & 87.5 & 23.3 & 57.2 & 15.2 \\ \hline \end{array}$$

a. Estimate the model Healthy $=\beta_0+\beta_1$ Fruits/ Vegetables $+\beta_2$ Exercise $+\beta_3$ Smoke $+\varepsilon$.

b. Analyze the data to determine if multicollinearity and changing variability are present.

Using data from 50 workers, a researcher estimates Wage $=\beta_0+\beta_1$ Education $+$ $\beta_2$ Experience $+\beta_3$ Age $+\varepsilon$, where Wage is the hourly wage rate and Education, Experience, and Age are the years of higher education, the years of experience, and the age of the worker, respectively. A portion of the regression results is shown in the following table.

$$\begin{array}{|l|c|c|c|c|} \hline & \text { Coefiicients } & \text { Standard Error } & \text { tStat } & \text { p-value } \\ \hline \text { Intercept } & 7.87 & 4.09 & 1.93 & 0.0603 \\ \hline \text { Education } & 1.44 & 0.34 & 4.24 & 0.0001 \\ \hline \text { Experience } & 0.45 & 0.14 & 3.16 & 0.0028 \\ \hline \text { Age } & -0.01 & 0.08 & -0.14 & 0.8920 \\ \hline \end{array}$$

a. Interpret the estimates for $\beta_1$ and $\beta_2$.

b. What is the sample regression equation?

c. Predict the hourly wage rate for a 30 -year-old worker with 4 years of higher education and 3 years of experience.

In a simple linear regression, the following sample regression equation is obtained:

$$\hat{y}=436-17 x .$$

a. Interpret the slope coefficient.

b. Predict $y$ if $x$ equals $-15$.

Healthy living has always been an important goal for any society. In an ad campaign for Walt Disney, former First Lady Michelle Obama shows parents and children that eating well and exercising can also be fun (USA Today, September 30,2010 ). Consider a regression model that conjectures that fruits and vegetables and regular exercising have a positive effect on health and smoking has a negative effect on health. The sample consists of the percentage of these variables observed in various states in the United States in 2009. A portion of the data is shown in the accompanying table.

Use the given data to construct a boxplot and identify the 5-number summary. Listed below are the measured radiation absorption rates (in W / kg) corresponding to these cell phones: iPhone 5S, BlackBerry Z30, Sanyo Vero, Optimus V, Droid Razr, Nokia N97, Samsung Vibrant, Sony Z750a, Kyocera Kona, LG 02, and Virgin Mobile Supreme. The data are from the Federal Communications Commission.

$$\begin{array} { c c c c c c c c c c } { 1.18 } & { 1.41 } & { 1.49 } & { 1.04 } & { 1.04 } & { 0.74 } & { 0.89 } & { 1.42 } & { 1.45 } & { 0.51 } & { 1.38 } \end{array}$$

In a simple linear regression based on 30 observations, it is found that $S S E=2540$ and $S S T=13,870$.

a. Calculate $s_e^2$ and $s_e$.

b. Calculate $R^2$.

The accompanying data file shows the square footage and associated property taxes for 20 homes in an affluent suburb 30 miles outside New York City.

c. What proportion of the sample variation in property taxes is unexplained by the home's size?

A social scientist would like to analyze the relationship between educational attainment (in years of higher education) and annual salary (in $\$ 1,000$ s). He collects data on 20 individuals. A portion of the data is as follows:

$$\begin{array}{|c|c|} \hline \text { Salary } & \text { Education } \\ \hline 40 & 3 \\ \hline 53 & 4 \\ \hline \vdots & \vdots \\ \hline 38 & 0 \\ \hline \end{array}$$

What is the predicted salary for an individual who completed $7$ years of higher education?

The following table shows a portion of the monthly returns data (in percent) for 2010-2016 for two of Vanguard's mutual funds: the Vanguard Energy Fund and the Vanguard Healthcare Fund.

$$\begin{array}{|c|c|c|} \hline \text { Date } & \text { Energy } & \text { Healthcare } \\ \hline \text { Jan-10 } & -4.86 & -0.13 \\ \hline \text { Feb-10 } & 1.50 & 0.58 \\ \hline \vdots & \vdots & \vdots \\ \hline \text { Dec-16 } & -0.30 & -5.26 \\ \hline \end{array}$$

c. At the $5 \%$ significance level, what is the conclusion to the test? Are the returns on the mutual funds correlated?

Consider the following competing hypotheses and accompanying sample data.

$$\begin{aligned} &H_0: \rho_S \leq 0 \\ &H_A: \rho_S>0 \end{aligned} \quad r_S=0.64 \text { and } n=50$$

b. At the 1% significance level, what is the conclusion?

Consider the following competing hypotheses and accompanying sample data.

$$H_0: p_1-p_2 \geq 0$$

$$H_A: p_1-p_2<0$$

$$x_1=250 \quad x_2=275$$

$$n_1=400 \quad n_2=400$$

b. Find the $p$-value.

The following table shows a portion of the monthly returns data (in percent) for 2010–2016 for two of Vanguard’s mutual funds: the Vanguard Energy Fund and the Vanguard Healthcare Fund.

$$\begin{array}{|c|c|c|} \hline \text { Date } & \text { Energy } & \text { Healthcare } \\ \hline \text { Jan-10 } & -4.86 & -0.13 \\ \hline \text { Feb-10 } & 1.50 & 0.58 \\ \hline \vdots & \vdots & \vdots \\ \hline \text { Dec-16 } & -0.30 & -5.26 \\ \hline \end{array}$$