Exercise Answers

Answers to Chapter 1 Exercises

1. [latex]s_{1}^2 = 125, s_{2}^2=17[/latex]
2. The larger the variance, the more spread the data.
3. 69<x<81
4. 85<x<115, max value 115

Answers to Chapter 2 Exercises

1. 4,200 choices
2. 35,152 codes
3. 32,760 ways
4. a. 35,254,642,500 ways, b. 21,646,947,168,000 ways, c. 65,536 ways
5. [latex]\frac{1}{30}[/latex]
6. [latex]\frac{1}{33}[/latex]
7. [latex]\frac{12}{13}[/latex]
8. [latex]\frac{23}{38}[/latex]
9. [latex]\frac{1}{8}[/latex]
10. [latex]\frac{91}{990}[/latex]
11. [latex]\frac{4}{5}[/latex]
12. a. [latex]\frac{2}{5}[/latex], b. [latex]\frac{71}{150}[/latex], c. [latex]\frac{37}{50}[/latex], d. [latex]\frac{3}{10}[/latex], e. [latex]\frac{43}{75}[/latex]

Answers to Chapter 3 Exercises

1. a. discrete, b. continuous
2. [latex]p_k>0,\ \ \ sump_k=1[/latex]
3. [latex]f\left(x\right)\geq0,\ \ \ \int_{-\infty}^{\infty}f\left(x\right)dx=1[/latex]
4. a.

Valid Probability Distribution

b.

Not a Valid Probability Distribution

1.  a. [latex]p\left(3\right)=0.1[/latex], b. x=2, c. 0.62, d. 0.88

1. a. 1.6, b. 1.05
2. a. 0.00078, b. 2
3. a. binomial; 0.39, b. not a binomial; probability of success is not the same for each trial. (answers may vary)
4. a. binomial, b. 72.8, , 6.552, 2.560 c. 0.120, d. 0.999[latex]\approx1[/latex], e. [latex]\approx0[/latex]

Answers to Chapter 4 Exercises

1. a. [latex]3e^{-3x}[/latex], b. 0.988, c. 1, d. [latex]\Gamma(1)[/latex], e. 0.9985 hrs after 11am (ie. by 12pm)
2. a. 0.5, b. 0
3. a. [latex]y=x^2+2x+3[/latex], b. [latex]y=\frac{3}{1390}\left(x^2+2x+3\right)[/latex]
4. a. [latex]\left\{\begin{array}{11}\frac{3}{8}x^2,\ \ \ 0\leq x\leq2 \\0\ \ \ \ \ \ \ \ \ elsewhere \\ \end{array}\right.[/latex], b. 0.125, c. 1.5, d. 0.15
5. a. 0.5, b. 0.05

Answers to Chapter 5 Exercises

1. [latex]\lim_{n\to\infty} {\frac{\sigma}{\sqrt{n}}}=0[/latex]
2. a. [latex]\int_{-z_{0.1}}^{z_{0.1}}{\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}}dx=0.8[/latex], b. [latex]\int_{-z_{0.025}}^{z_{0.025}}{\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}}dx=0.95[/latex], c. [latex]\int_{-z_{0.005}}^{z_{0.005}}{\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}}dx=0.99[/latex]
3. Yes, there is a statistically significant difference.
4. a. 3, b. 3
5. 0.28868
6. a. 2, b. 2
7. a. [latex]\frac{1}{7}[/latex], b. [latex]\frac{1}{49}[/latex], c. [latex]\frac{7}{7-t};\ \frac{1}{7};\ \frac{2}{49}[/latex], d. [latex]\frac{1}{49}[/latex], e. [latex]\frac{2}{49}[/latex]
8. a. [latex]\frac{3}{4}[/latex], b. [latex]\frac{1}{3t}\left(e^{3t}-1\right)[/latex]; [latex]\frac{3}{2}[/latex]; 3 , c. [latex]\frac{3}{4}[/latex]