9.7 ${\beta }_1=1/3,{\beta }_0=14/3,y=14/3+(1/3)x$

9.11

1. a. dependent: size; independent: distance

2. c. $y={\beta }_0+{\beta }_{1}x+\epsilon$

9.13

1. a. dependent: ratio; independent: diameter

2. c. $y={\beta }_0+{\beta }_{1}x+\epsilon$

9.15 difference between the observed and predicted

9.17 true

9.19

1. b. $\widehat{y}=7.10-.78x$

9.21

1. c. ${\widehat{\beta }}_1=.918,{\widehat{\beta }}_0=.020$

2. e.$-1\text{to}7$

9.23

1. a. yes; positive

2. d. $\widehat{y}=320.64+.084x$

3. e. 3.565

9.25

1. $y={\beta }_0+{\beta }_{1}x+\epsilon$

2. $\widehat{y}=19.393-8.036x$

3. y-intercept: when $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}=0$, predicted wicking $\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}=19.393\mathrm{ m}\mathrm{m}$ ; slope: for every 1-unit increase in concentration, wicking length decreases 8.036 mm

9.27

1. positive

9.29

1. a. $y={\beta }_0+{\beta }_{1}x+\epsilon$

2. b. $\widehat{y}=250.14-.627x$

3. e. slope

9.31

1. $y={\beta }_0+{\beta }_{1}x+\epsilon$

2. positive

3. ${\widehat{\beta }}_1=210.8$ : for every additional resonance, frequency increases 210.8; ${\widehat{\beta }}_0=.020$ : no practical interpretation

9.33

1. $\widehat{y}=86.0-.260x$

2. yes

3. positive trend for females; positive trend for males

4. $\widehat{y}=39.3+.493x$ ; for every 1-inch increase in height for females, ideal partner’s height increases .493 inch

5. $\widehat{y}=23.3+.596x$ ; for every 1-inch increase in height for males, ideal partner’s height increases .596 inch

6. yes

9.35 yes, $\widehat{y}=5.22-.114x$ ; decrease by .114 pound

9.37 (1) $\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}=0$, (2) error variance is constant for all x, (3) errors are normally distributed, (4) errors are independent

9.39

1. 57.5; 3.194

2. 257.5; 6.776

3. 9.288; 1.161

9.41 $11.18:\mathrm{S}\mathrm{S}\mathrm{E}=1.22,s^2=.244,s=.494;$ $11.21:\mathrm{S}\mathrm{S}\mathrm{E}=5.135,{\mathrm{s}}^2=1.03,\mathrm{s}=1.01$

9.43 $s=.57;\approx 95\mathrm{\%}$ of angular sizes fall within 1.14 pixels of their respective predicted values

9.45

1. $\mathrm{S}\mathrm{S}\mathrm{E}=22.268,s^2=5.567,s=2.3594$

2. $\approx 95\mathrm{\%}$ of wicking length values fall within 4.72 mm of their respective predicted values

9.47

1. $\text{SSE}=2760,{\text{s}}^2=306.6,\text{and}\text{s}=17.51$

9.49

1. 5.37

2. 3.42

3. reading score

9.51

1. $\widehat{y}=23.3-.596x;s=2.06$

2. $\widehat{y}=39.3+.493x;s=2.32$

3. males

9.53 0

9.55 divide the value in half

9.57

1. $95\mathrm{\%}:31\pm 1.13;90\mathrm{\%}:31\pm .92$

2. $95\mathrm{\%}:64\pm 4.28;$ $90\mathrm{\%}:64\pm 3.53$

3. $95\mathrm{\%}:-.84\pm .67;90\mathrm{\%}:-.84\pm .55$

9.59

1. b. $\widehat{y}=2.554+.246x$

2. d. $t=.627$

3. e. fail to reject $H_0$

4. f. $.246\pm 1.81$

9.61 yes, $t=14.87$ ; (.0041, .0055)

9.63

1. negative linear trend

2. $\widehat{y}=9,658.24-171.573x$ ; for each 1-unit increase in search frequency, the total catch is estimated to decrease by 171.573 kg

3. $H_0:{\beta }_1=0,H_{\text{a}}:{\beta }_1<0$

4. $.0402/2=.0201$

5. reject $H_0$

9.65

1. $H_0:{\beta }_1=0,$ $H_{\text{a}}:{\beta }_1<0$

2. $t=-13.23,p\text{-value}=0$

3. reject $H_0$

9.67 $-.0023\pm .0019$ ; 95% confident that change in sweetness index for each 1-unit change in pectin is between $-.0042$ and $-.0004$

9.69

1. a. $y={\beta }_0+{\beta }_{1}x+\epsilon$

2. b.$\widehat{y}=-8.524+1.665x$

3. d. yes, $t=7.25$

4. e. $1.67\pm .46$

9.71

1. 95% confident that for each 1% increase in body mass, the percentage change in eye mass increases between 0.25% and 0.30%

2. 95% confident that for each 1% increase in body mass, the percentage change in orbit axis angle decreases between 0.05% and 0.14%.

9.73 $(-7.83,3.71)$

9.75

1. ${\widehat{\beta }}_0=.515,{\widehat{\beta }}_1=.000021$

2. yes

3. very influential

4. ${\widehat{\beta }}_0=.515,{\widehat{\beta }}_1=.000020,$ $p\text{-value}=.332$, fail to reject $H_0$

9.77 true

9.79

1. perfect positive linear

2. perfect negative linear

3. no linear

4. strong positive linear

5. weak positive linear

6. strong negative linear

9.81

1. $r=.985,r^2=.971$

2. $r=-.993,r^2=.987$

3. $r=0,r^2=0$

4. $r=0,r^2=0$

9.83 .877

9.85

1. weak, negative linear relationship between the first letter of the last name and response time

2. reject $H_0:\rho =0$

3. yes

9.87

1. 18% of sample variation in points scored can be explained by the linear model

2. $-.424$

9.89

1. a. moderate positive linear relationship; not significantly different from 0 at $\alpha =.05$

2. c. weak negative linear relationship; not significantly different from 0 at $\alpha =.10$

9.91

1. b. piano: $r^2=.1998$ ; bench: $r^2=.0032$ ; motorbike: $r^2=.3832$, armchair: $r^2=.0864$ ; teapot: $r^2=.9006$

2. c.Reject $H_0$ for all objects except bench and armchair

9.93

1. $H_0:{\beta }_1=0,H_{\text{a}}:{\beta }_1<0$

2. reject $H_0$

3. no; be careful not to infer a causal relationship

9.95 $r=.570,r^2=.325$

9.97

1. $r^2=0.4756$ ; 47.56% of the sample variation in molecular variance can be explained by altitude in the linear model

2. $r^2=0.5236$ ; 52.36% of the sample variation in molecular variance can be explained by population size in the linear model

9.99 E(y) represents mean of y for all experimental units with same x-value

9.101 true

9.103

1. $\widehat{y}=1.375+.875x$

2. 1.5

3. .1875

4. $3.56\pm .33$

5. $4.88\pm 1.06$

9.105

1. c. $4.65\pm 1.12$

2. d. $2.28\pm .63;-.414\pm 1.717$

9.107

1. Find a prediction interval for y when $x=10$

2. Find a confidence interval for E(y) when $x=10$

9.109 (92.298, 125.104)

9.111 run 1: 90% confident that for all runs with a pectin value of 220, mean sweetness index will fall between 5.65 and 5.84

9.113

1. 95% confidence interval for E(y)

2. (48.4, 64.6)

3. 95% confident that the mean in-game heart rate of all top-level water polo players who have a maximal oxygen uptake of 150 VO₂max is between 48.4% and 64.6%.

9.115

1. (67.07, 76.41); 95% confident that ideal partner’s height is between 67.07 and 76.41 in. when a female’s height is 66 in.

2. (58.39, 66.83); 95% confident that ideal partner’s height is between 58.39 and 66.83 in. when a male’s height is 66 in.

3. males; 66 in. is outside range of male heights in sample

9.117

1. (2.955, 4.066)

2. (1.020, 6.000)

3. prediction interval; yes

9.119

1. Brand A: $3.35\pm .59$ ; Brand B: $4.46\pm .30$

2. Brand A: $3.35\pm 2.22$ ; Brand B: $4.46\pm 1.12$

3. $-.65\pm 3.61$

9.121

1. $\widehat{y}=1.411+349x$

2. $t=1.05$, fail to reject $H_0:{\beta }_1=0;r^2=0.1219$ ; no

9.123 yes; $\widehat{y}=7.77+.000113x,t=4.04,\text{reject }{H}_0:{\beta }_1=0$

9.125 $-1$ ; 1

9.127

1. .4

2. $-.9$

3. $-.2$

4. .2

9.129

1. c. $r_s=.95$

2. d. $\text{reject}{H}_0$

9.131

1. $r_s=.934$

2. yes,

3. $p\text{-value}=.00035$

9.133

1. c. .713

2. d. $\text{| }{r}_s|>.425$

3. e. $\text{reject}{H}_0$

9.135 Private: do not reject $H_0,p\text{-value}=.103$ ; Public: reject $H_0,p\text{-value}=.002$

9.137 yes

9.139 $r_s=.341$ reject $H_0,p-\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}=.0145$

9.141 $E(y)={\beta }_0+{\beta }_{1}x$

9.143 true

9.145

1. b. $\widehat{y}=x;\widehat{y}=3$

2. c. $\widehat{y}=x$

3. d. least squares line has the smallest SSE

9.149

1. b. .185

9.151

1. a. positive

2. b. yes

3. c. $\widehat{y}=-58.86+554.5x$

4. e. slope: for each 01-point increase in batting average, estimated number of games won will increase by 5.54

5. f. $t=2.81$, reject $H_0:{\beta }_1=0$

6. g. $.396;\approx 40\mathrm{\%}$ of sample variation in games won is explained by the linear model

7. h. yes

9.153

1. a. $y={\beta }_0+{\beta }_{1}x+\epsilon$

2. b. $\widehat{y}=175.70-.8195x$

3. e. $t=-3.43$, reject $H_0$

9.155

1. $\widehat{y}=6.31+0.967x$

2. no practical interpretation

3. For every 1 micrometer increase in mean pore diameter, porosity is estimated to increase by 0.967%

4. $t=2.68$, reject $H_0$

5. $r=.802,r^2=.643$

6. (8.56, 23.40)

9.157

1. yes

2. ${\widehat{\beta }}_0=-3.05,{\widehat{\beta }}_1=.108$

3. $.t=4.00$, reject $H_0$

4. $r=.756,r^2=.572$

5. 1.09

6. yes

9.159

1. ${\widehat{\beta }}_0=-13.49,{\widehat{\beta }}_1=-.0528$

2. $-.0528\pm .0178$ ; yes

3. $r^2=.854$

4. (.5987, 1.2653)

9.161

1. $\widehat{y}=46.4x$

2. $\widehat{y}=478.44+45.15x$

3. no, $t=.91$

9.163 $\widehat{y}=2.55+2.76x;t=12.66\text{reject }{H}_0$