13.99.1

$H_0:{\beta }_1=0$: There is no linear relationship between Education ($x$) and Annual Earnings ($y$). $H_a:{\beta }_1\ne 0$: There is a linear relationship between Education ($x$) and Annual Earnings ($y$). Reject $H$ if $t_{\text{data}}\ge 2.571$ or $t_{\text{data}}\le -2.571$ . Since $t_{\text{data}}=7.542\ge 2.571$, we reject $H_0$. There is evidence at level of significance $\alpha =0.05$ that ${\beta }_1\ne 0$ and that there is a linear relationship between Education ($x$) and Annual Earnings ($y$).

13.99.3

$H_0:{\beta }_1=0$: There is no linear relationship between Age ($x$) and Price $H_a:{\beta }_1\ne 0$: There is a linear relationship between Age ($x$) and Price ($y$). Reject $H_0$ if $t_{\text{data}}\ge 2.306$ or $t_{\text{data}}\le -2.306$. Since $t_{\text{data}}=-15.0124\le -2.306$, we reject $H_0$. There is evidence at level of significance $\alpha =0.05$ that ${\beta }_1\ne 0$ and that there is a linear relationship between Age ($x$) and Price ($y$).

13.99.5

(0.339, 1.029). We are 95% confident that the interval (0.339, 1.029) captures the population slope $b_1$ of the relationship between high school GPA and first-year college GPA.

13.99.7

(a) 20.92 thousand dollars (b) (14.28, 27.56). We are 95% confident that the mean annual salary for people with 10 years of education lies between 14.28 thousand dollars and 27.56 thousand dollars. (c) (6.55, 35.29). We are 95% confident that the annual salary for a randomly selected person with 10 years of education lies between 6.55 thousand dollars and 35.29 thousand dollars.

13.99.9

(a) 6.818 thousand dollars (b) (5.805, 7.831). We are 95% confident that the mean price for 8-year-old cars of this make and model lies between 5.805 thousand dollars and 7.831 thousand dollars (c) (4.902, 8.733). We are 95% confident that the price for a randomly selected 8-year-old car of this make and model lies between 4.902 thousand dollars and 8.733 thousand dollars.

13.99.11

(a) Test $1:H_0:{\beta }_1=0$: There is no linear relationship between $y$ and $x_1$. $H_a:{\beta }_1\ne 0$: There is a linear relationship between $y$ and $x_1$. Reject $H_0$ if the $p$-value $\le \alpha =0.05$. Test $2:H_0:{\beta }_2=0$: There is no linear relationship between $y$ and $x_2$. $H_a:{\beta }_2\ne 0$: There is a linear relationship between $y$ and $x_2$. Reject $H_0$ if the $p$-value $\le \alpha =0.05$. Test $3:H_0:{\beta }_3=0$: There is no linear relationship between $y$ and $x_3$. $H_a:{\beta }_3\ne 0$: There is a linear relationship between $y$ and $x_3$. Reject $H_0$ if the $p$-value $\le \alpha =0.05$. (b) Test $1:t=1.86$, with $p\text{-value}=0.112$; Test $2:t=-0.27$, with $p\text{-value}=0.798$; Test 3: $t=-1.07$, with $p\text{-value}=0.326$. (c) Test 1: The $p\text{-value}=0.112$, which is not $\le \alpha =0.05$. Therefore we do not reject $H_0$. There is insufficient evidence of a linear relationship between $y$ and $x_1$. Test 2: The $p\text{-value}=0.798$, which is not $\le \alpha =0.05$. Therefore we do not reject $H_0$. There is insufficient evidence of a linear relationship between $y$ and $x_2$. Test 3: The $p\text{-value}=0.326$, which is not $\le \alpha =0.05$. Therefore we do not reject $H_0$. There is insufficient evidence of a linear relationship between $y$ and $x_3$.

13.99.13

The preceding scatterplot of the residuals versus fitted values shows no strong evidence of unhealthy patterns. Thus, the independence assumption, the constant variance assumption, and the zero-mean assumption are verified. Also, the normal probability plot of the residuals indicates no evidence of departure from normality of the residuals. Therefore we conclude that the regression assumptions are verified.