13.1.1

The regression equation is calculated from a sample and is valid only for values of $x$ in the range of the sample data. The population regression equation may be used to approximate the relationship between the predictor variable $x$ and the response variable $y$ for the entire population of ($x$, $y$) pairs.

13.1.3

We construct a scatterplot of the residuals against the fitted values and a normal probability plot of the residuals. We must make sure that the scatterplot contains no strong evidence of any unhealthy patterns and that the normal probability plot indicates no evidence of departures from normality in residuals.

13.1.5

There is no relationship between $x$ and $y$.

13.1.7

(a) and (b)

(c) and (d) See Solutions Manual. (e) The scatterplot of the residuals contains an unhealthy pattern, so the regression assumptions are not verified.

13.1.9

(a) and (b)

(c) and (d) See Solutions Manual. (e) The scatterplot of the residuals contains an unhealthy pattern, so the regression assumptions are not verified.

13.1.11

(a) and (b)

(c) and (d) See Solutions Manual. (e) The scatterplot of the residuals contains an unhealthy pattern, so the regression assumptions are not verified.

13.1.13

(a) and (b)

(c)

(d)

(e) The scatterplot in (c) 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 in (d) indicates no evidence of departure from normality of the residuals. Therefore we conclude that the regression assumptions are verified.

13.1.15

(a) $t_{\text{crit}}=3.182$ (b) $s=1.58113883$ (c) ${\displaystyle \sum {(x-\overline{x})}^2}=10$ (d) $t_{\text{data}}=5$ (e) $H_0:{\beta }_1=0$: There is no linear relationship between $x$ and $y$. $H_a:{\beta }_1\ne 0$: There is a linear relationship between $x$ and $y$. Reject $H_0$ if $t_{\text{data}}\ge 3.182$ or $t_{\text{data}}\le -3.182$. Since $t_{\text{data}}=5\ge 3.182$, 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 $x$ and $y$.

13.1.17

(a) $t_{\text{crit}}=3.182$ (b) $s=2.529822128$ (c) ${\displaystyle \sum {(x-\overline{x})}^2}=10$ (d) $t_{\text{data}}=5$ (e) $H_0:{\beta }_1=0$: There is no linear relationship between $x$ and $y$. $H_a:{\beta }_1\ne 0$: There is a linear relationship between $x$ and $y$. Reject $H_0$ if $t_{\text{data}}\ge -3.182$ or $t_{\text{data}}\le -3.182$. Since $t_{\text{data}}=5\ge -3.182$, 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 $x$ and $y$.

13.1.19

(a) $s=2.581988897$ (b) ${\displaystyle \sum {(x-\overline{x})}^2}=100$ (c) $t_{\text{data}}=-6.1237$ (d) $p\text{-value}=0.0088$ (e) $H_0:{\beta }_1=0$: There is no linear relationship between $x$ and $y$. $H_a:{\beta }_1\ne 0$: There is a linear relationship between $x$ and $y$. Reject $H_0$ if $p$-value $\le 0.05$. Since $p$-value $=0.0088\le 0.05$, 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 $x$ and $y$.

13.1.21

(a) $s=0.3651483717$ (b) ${\displaystyle \sum {(x-\overline{x})}^2}=10$ (c) $t_{\text{data}}=5.1962$

(d) $p\text{-value}=0.0138$ (e) $H_0:{\beta }_1=0$: There is no linear relationship between $x$ and $y$. $H_a:{\beta }_1\ne 0$: There is a linear relationship between $x$ and $y$. Reject $H_0$ if $p$-value $\le 0.05$. Since $p$-value $=0.0138\le 0.05$, 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 $x$ and $y$.

13.1.23

(a) $t_{\alpha /2}=3.182$ (b) $E=1.591$ (c) (0.909, 4.091)

13.1.25

(a) $t_{\alpha /2}=3.182$ (b) $E=2.5456$ (c) (1.4544, 6.5456)

13.1.27

(a) $t_{\alpha /2}=3.182$ (b) $E=0.2598$ (c) (–0.7598, −0.2402)

13.1.29

(a) $t_{\alpha /2}=3.182$ (b) $E=0.3674$ (c) (0.2326, 0.9674). TI-83/84: (0.2325, 0.9675)

13.1.31

$H_0:{\beta }_1=0$: There is no linear relationship between $x$ and $y$. $H_a:{\beta }_1\ne 0$: There is a linear relationship between $x$ and $y$. Since the confidence interval from Exercise 23 (c) does not contain zero, we may conclude that ${\beta }_1\ne 0$ and that a linear relationship exists between $x$ and $y$, at level of significance $\alpha =0.05$.

13.1.33

$H_0:{\beta }_1=0$: There is no linear relationship between $x$ and $y$. $H_a:{\beta }_1\ne 0$: There is a linear relationship between $x$ and $y$. Since the confidence interval from Exercise 25 (c) does not contain zero, we may conclude that ${\beta }_1\ne 0$ and that a linear relationship exists between $x$ and $y$, at level of significance $\alpha =0.05$.

13.1.35

$H_0:{\beta }_1=0$: There is no linear relationship between $x$ and $y$. $H_a:{\beta }_1\ne 0$ There is a linear relationship between $x$ and $y$. Since the confidence interval from Exercise 27 (c) does not contain zero, we may conclude that ${\beta }_1\ne 0$ and that a linear relationship exists between $x$ and $y$, at level of significance $\alpha =0.05$.

13.1.37

$H_0:{\beta }_1=0$: There is no linear relationship between $x$ and $y$. $H_a:{\beta }_1\ne 0$ There is a linear relationship between $x$ and $y$. Since the confidence interval from Exercise 29 (c) does not contain zero, we may conclude that ${\beta }_1\ne 0$ and that a linear relationship exists between $x$ and $y$, at level of significance $\alpha =0.05$.

13.1.39

$H_0:{\beta }_1=0$: There is no relationship between volume ($x$) and weight ($y$). $H_a:{\beta }_1\ne 0$: There is a linear relationship between volume ($x$) and weight ($y$). Reject $H_0$ if the $p$-value ≤ 0.05. Since the $p$-value $\approx 0.0006$ is $\le 0.05$, we reject $H_0$. There is evidence for a linear relationship between volume ($x$) and weight ($y$).

13.1.41

$H_0:{\beta }_1=0$: There is no relationship between Low ($x$) and High ($y$).

$H_a:{\beta }_1\ne 0$: There is a linear relationship between Low ($x$) and High ($y$).

Reject $H_0$ if $t_{\text{data}}\ge 2.776$. Since $t_{\text{data}}\approx 12.09$ is $\ge 2.776$, we reject $H_0$. There is evidence for a linear relationship between Low ($x$) and High ($y$).

13.1.43

$H_0:{\beta }_1=0$: No linear relationship exists between DJIA and darts.

$H_a:{\beta }_1\ne 0$: A linear relationship exists between DJIA and darts.

Reject $H_0$ if the $p$-value $\le \alpha =0.05$. $t_{\text{data}}\approx 2.1749$. $p$-value $=0.0725703888$.

The $p\text{-value}=0.0725703888$ is not ≤ $\alpha =0.05$, so we do not reject $H_0$. Insufficient evidence exists, at level of significance $\alpha =0.05$, for a linear relationship between DJIA and darts.

13.1.45

$H_0:{\beta }_1=0$: No linear relationship exists between age and shots.

$H_a:{\beta }_1\ne 0$: A linear relationship exists between age and shots.

Reject $H_0$ if the $p$-value $\le \alpha =0.05$. $t_{\text{data}}\approx 0.1817$. $p\text{-value}=0.8603048707$.

The $p\text{-value}=0.8603048707$ is not $\le \alpha =0.05$, so we do not reject $H_0$. insufficient evidence exists, at level of significance $\alpha =0.05$, for a linear relationship between a patient's age and the number of shots taken by the patient.

13.1.47

(a) $E=0.3312$ (b) (1.2688, 1.9312) (c) We are 95% confident that the interval (1.2688, 1.9312) captures the population slope ${\beta }_1$ of the relationship between volume and weight.

13.1.49

(a) $E=0.2402$ (b) (0.8064, 1.2868). TI-83/84: (0.8063, 1.2868) (c) We are 95% confident that the interval (0.8063, 1.2868) captures the population slope ${\beta }_1$ of the relationship between Low and High.

13.1.51

(a) 1.5910 (b) (–0.1769, 3.0051) (c) We are 95% confident that the interval (–0.1769, 3.0051) captures the slope ${\beta }_1$ of the population regression line. That is, we are 95% confident that for each additional $1 the stocks in the DJIA gain in one day, the daily change in the stocks in the portfolio predicted by the darts lies between –$0.039 and $0.66248.

13.1.53

(a) 0.19824 shot (b) (–0.1826, 0.21388) (c) We are 95% confident that the interval (–0.1826, 0.21388) captures the slope ${\beta }_1$ of the population regression line. That is, we are 95% confident that for each additional year in a patient's age, the change in the number of shots taken by the patient lies between −0.1826 shot and 0.21388 shot.

13.1.55

(a) (–6.647, 49.817) (b) 0 lies in the interval, so we do not reject $H_0$.

13.1.57

The scatterplot of the residuals versus the fitted values shows no evidence of the unhealthy patterns shown in Figure 4. 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 departures from normality in the residuals. Therefore, we conclude that the regression assumptions are verified and it is okay to proceed with the regression.

13.1.59

22.51 runs; by far the highest number of runs scored but the third highest batting average.

13.1.61

Against; the points appear to lie near a line with a positive slope.

13.1.63

$H_0:{\beta }_1=0$: There is no linear relationship between SAT Reading score ($x$) and SAT Math score ($y$). $H_a:{\beta }_1\ne 0$: There is a linear relationship between SAT Reading score ($x$) and SAT Math score ($y$). Reject $H_0$ if $t_{\text{data}}\ge 2.353$ or $t_{\text{data}}\le -2.353$. Since $t_{\text{data}}=3.1963\ge -2.353$, we reject $H_0$. There is evidence at level of significance $\alpha =1.10$ that ${\beta }_1\ne 0$ and that there is a linear relationship between SAT Reading score ($x$) and SAT Math score ($y$).

13.1.65

Yes. Since the confidence interval from Exercise 64 does not contain zero, we may conclude that ${\beta }_1\ne 0$ and that a linear relationship exists between SAT Reading score ($x$) and SAT Math score ($y$), at level of significance $\alpha =0.10$.

13.1.67

(a) $t_{\text{data}}$ increases if $b_1$ is positive and decreases if $b_1$ is negative. (b) $r^2$ remains the same. (c) $s$ decreases. (d) $p$-value decreases. (e) Since we don't know what the new $p$-value will be, we don't know if the $p$-value will decrease enough to change the conclusion from “Do not reject $H_0$” to “Reject $H_0$.”

13.1.69

(a) $t_{\text{data}}$ increases if $b_1$ is positive and decreases if $b_1$ is negative.

(b) $r^2$ increases. (c) $s$ decreases. (d) $p$-value decreases. (e) Unchanged.

13.1.71

(a-b) Decrease (c-d) Increase (e) Depends on the new $p$-value.

13.1.73

(a) The scatterplot of the residuals contains evidence of an unhealthy pattern and the normal probability plot indicates evidence of departures from normality in the residuals. Therefore we conclude that the regression assumptions are not verified. (b) (7.821, 8.437). We are 95% confident that the interval (13.5483, 14.6201) captures the population slope $b_1$ of the relationship between fat per gram and calories per gram. (c) Yes (d) $H_0:{\beta }_1=0$. There is no relationship between fat per gram ($x$) and calories per gram $H_0:{\beta }_1=0$. There is a linear relationship between fat per gram ($x$) and calories per gram ($y$). Reject $H_0$ if $p\text{-value}\le 0.05$. $t_{\text{data}}=51.84$ $p$-value ≈ 0. Since the $\le 0.05$, we reject $H_0$. There is evidence for a linear relationship between fat per gram ($x$) and calories per gram ($y$).

13.1.75

(a) No (b) Positive relationship (c) Unclear

13.1.77

(a) No (b) No apparent relationship between the variables (c) The weight of vehicles is the predictor variable and the severity of the leg injuries should be the response variable.