5.8: Chapter 5 Solutions
- Page ID
- 79037
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. a. \(10\)
b. 1.25
3. N(10, 1.25)
5. 0.7799
7. 1.675
9. Mean = 25, standard deviation = 0.29
10. Mean = 48, standard deviation = 0.83
11. Mean = 90, standard deviation = 0.75
12. Mean = 120, standard deviation = 0.38
13. Mean = 17, standard deviation = 0.17
14. Expected value = 17, standard deviation = 0.05
15. Expected value = 38, standard deviation = 0.43
16. Expected value = 14, standard deviation = 0.65
17. 0.9818
18. 0.8788
19. 0.3050
20. 0.8294
21. 0.4884
22. 0.5000
23. 0.4761
24. 0.5160
25.
- \(X\) = amount of change students carry
- \(X \sim N(0.88, 0.31)\)
- \(\overline x\) = average amount of change carried by a sample of 25 students.
- \(\overline x \sim N(0.88, 0.062)\)
- \(0.2543\)
- \(0.8753\)
- The distributions are different. Part 1 is normal for individual observations and part 2 is normal for sample means.
27.
- length of time for an individual to complete \(IRS\) form 1040, in hours.
- mean length of time for a sample of 36 taxpayers to complete \(IRS\) form 1040, in hours.
- \(N(10.53, \frac{1}{3})\)
- Yes. I would be surprised, because the probability is almost 0.
- No. I would not be totally surprised because the probability is 0.2296.
29.
- the length of a song, in minutes, in the collection
- \(N(2, 0.43)\)
- the average length, in minutes, of the songs from a sample of five albums from the collection
- \(N(2.75, 0.066)\)
- 2.706 minutes
- 0.088 minutes
31.
- True. The mean of a sampling distribution of the means is approximately the mean of the data distribution.
- True. According to the Central Limit Theorem, the larger the sample, the closer the sampling distribution of the means becomes normal.
- The standard deviation of the sampling distribution of the means will decrease as the sample size increases.
33.
- \(X\) = the yearly income of someone in a third world country
- the average salary from samples of 1,000 residents of a third world country
- \(\overline x \sim N\left(2000, \frac{8000}{\sqrt{1000}}\right)\)
- Very wide differences in data values can have averages smaller than standard deviations.
- The distribution of the sample mean will have higher probabilities closer to the population mean.
\(P(2000 < \overline x < 2100) = 0.1537 \)
\(P(2100 < \overline x < 2200) = 0.1317\)
35. b
36. 64
37.
- Yes
- Yes
- Yes
- 0.6
38. 400
39. 2.5
40. 25
41. 0.955
42. 0.927
43. 0.648
44. 0.101
45. 0.273