9.8: Chapter 9 Solutions
- Page ID
- 79088
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)3. \(SS_{Factor}=4,939.2\)
5. \(df_{num}=2\)
7. \(MS_{Factor}=2,469.6\)
9. \(F_{obs}=3.742\)
11. \(F_{\alpha=.05, 2, 12}=3.89\). \(F_{obs}\) does not exceed \(F_{critical}\), so we fail to reject \(H_0\).
13. \(H_{0} : \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}\); \(H_a\): At least two of the means (\(\mu_{1}, \mu_{2}, \mu_{3}, \mu_{4}\)) are not equal.
15. \(df_{num}=3\)
17. \(SS_{Within}=13.2\)
19. \(MS_{Within}=.825\)
21.
| Source of variation | \(SS\) | \(df\) | \(MS\) | \(F\) |
|---|---|---|---|---|
| Between | 25.75 | 3 | 8.583 |
10.404 |
| Within | 13.2 | 16 |
.825 |
|
| Total | 38.95 | 19 |
Table \(\PageIndex{1}\)
23. The closest available \(F\)-value to our \(F_{obs}=10.404\) at (3, 16) \(df\) is 9.01, which corresponds to \(\alpha\) of .01. Therefore, the \(p\)-value must be less than .01. Because this \(p\)-value is less than \(\alpha\) = .05, we reject \(H_{0}\).
27. \(df_{num}=4\)
29. \(SS_{Factor}=195.6\); \(MS_{Factor}=48.9\)
31. \(F_{obs}=2.060\)
33. \(F_{\alpha=.05, 4, 10}=3.48\)
35. No, \(F_{obs}\) does not exceed \(F_{critical}\), and the approximate \(p\)-value exceeds \(\alpha\), so we fail to reject \(H_0\).
37. \(df_{num}=2\)
39. \(SS_{Between}=5,700.4\); \(MS_{Between}=2,850.2\)
41. \(F_{obs}=3.610\)
43. \(F_{\alpha=.01, 2, 12}=6.93\)
45. No, \(F_{obs}\) does not exceed \(F_{critical}\), and the approximate \(p\)-value exceeds \(\alpha\), so we fail to reject \(H_0\).
47. \(SS_{Between}=.903\)
49. \(df_{num}=4\)
51. \(F_{obs}=4.220\)
53.
a. Decision: Reject \(H_0\), because the approximate \(p\)-value (between .01 and .05) is less than \(\alpha\).
b. Conclusion: There is a significant difference in at least one of these regions of the country for the average age at which teenagers obtain their driver's licenses.
55. \(H_{0} : \mu_{East}=\mu_{Central}=\mu_{West}\); \(H_a\): At least two of the means (\(\mu_{East}, \mu_{Central}, \mu_{West}\)) are not equal.
\(F_{obs}=\frac{\frac{344.164}{2}}{\frac{1,219.550}{11}}=\frac{172.082}{110.868}=1.552\)
\(F_{\alpha=.05, 2, 12}=3.98\)
\(F_{obs}\) does not exceed \(F_{critical}\), so we fail to reject \(H_0\).
57. \(H_{0} : \mu_{Linda}=\mu_{Tuan}=\mu_{Javier}\); \(H_a\): At least two of the means (\(\mu_{Linda}, \mu_{Tuan}, \mu_{Javier}\)) are not equal.
\(F_{obs}=\frac{\frac{23.212}{2}}{\frac{208.324}{12}}=\frac{11.606}{17.360}=.669\)
\(F_{\alpha=.01, 2, 12}=6.93\)
\(F_{obs}\) does not exceed \(F_{critical}\), so we fail to reject \(H_0\).
59. \(H_{0} : \mu_{Home}=\mu_{News}=\mu_{Health}=\mu_{Computer}\); \(H_a\): At least two of the means (\(\mu_{Home}, \mu_{News}, \mu_{Health}, \mu_{Computers}\)) are not equal.
| Source of variation | \(SS\) | \(df\) | \(MS\) | \(F\) |
|---|---|---|---|---|
| Between | 34,288.6 | 3 | 11,429.533 |
8.689 |
| Within | 21,047.6 | 16 |
1,315.475 |
|
| Total | 55,336.2 | 19 |
Table \(\PageIndex{2}\)
\(F_{\alpha=.05, 3, 16}=3.24\). \(F_{obs}\) does exceed \(F_{critical}\), so we can reject \(H_0\).
61. \(H_{0} : \mu_{CNN}=\mu_{FOX}=\mu_{Local}\); \(H_a\): At least two of the means (\(\mu_{CNN}, \mu_{FOX}, \mu_{Local}\)) are not equal.
\(F_{obs}=\frac{\frac{1,967.925}{2}}{\frac{3,375.133}{14}}=\frac{983.963}{241.081}=4.081\)
\(F_{\alpha=.05, 2, 14}=3.74\)
\(F_{obs}\) exceeds \(F_{critical}\), so we reject \(H_0\).
63. \(H_{0} : \mu_{White}=\mu_{Black}=\mu_{Hispanic}=\mu_{Asian}\); \(H_a\): At least two of the means (\(\mu_{White}, \mu_{Black}, \mu_{Hispanic}, \mu_{Asian}\)) are not equal.
| Source of variation | \(SS\) | \(df\) | \(MS\) | \(F\) |
|---|---|---|---|---|
| Between | 13.032 | 3 | 4.344 |
.885 |
| Within | 73.600 | 15 |
4.907 |
|
| Total | 86.632 | 18 |
Table \(\PageIndex{3}\)
\(F_{\alpha=.01, 3, 15}=5.42\). \(F_{obs}\) doesn't exceed \(F_{critical}\), so we cannot reject \(H_0\).
65.
- Yes.
- Because \(n_1 = n_2 = n_3\).
- \(\bar{x}_{Heavy} = 4.55, s_{Heavy} = 1.00; \bar{x}_{Medium} = 4.525, s_{Medium} = 1.13; \bar{x}_{Light} = 2.60, s_{Light} = .70\)
-
- g = 3
- n = 12
- \(SS_{Between}\) = 10.012
- \(MS_{Between}\) = 5.006
- \(SS_{Within}\) = 8.277
- \(MS_{Within}\) = .920
- \(df_{num}\) = 2, \(df_{denom}\) = 9
- \(F\) statistic = 5.443
- \(F\)-critical = 8.02
- Graph: Check student's solution.
- Decision: \(F_{obs}\) doesn't exceed \(F_{critical}\), so we cannot reject \(H_0\).
- Conclusion: There is not sufficient evidence to conclude that the mean paper airplane flight distances differ based on paper weights.