## Answers to Chapter 1 Exercises

1. $s_{1}^2 = 125, s_{2}^2=17$
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. $\frac{1}{30}$
6. $\frac{1}{33}$
7. $\frac{12}{13}$
8. $\frac{23}{38}$
9. $\frac{1}{8}$
10. $\frac{91}{990}$
11. $\frac{4}{5}$
12. a. $\frac{2}{5}$, b. $\frac{71}{150}$, c. $\frac{37}{50}$, d. $\frac{3}{10}$, e. $\frac{43}{75}$

## Answers to Chapter 3 Exercises

1. a. discrete, b. continuous
2. $p_k>0,\ \ \ sump_k=1$
3. $f\left(x\right)\geq0,\ \ \ \int_{-\infty}^{\infty}f\left(x\right)dx=1$
4. a.
 x 1 2 3 5 p(x) 0.4 0.1 0.3 0.2

Valid Probability Distribution

b.

 x 0 1 2 3 p(x) 0.3 0.2 0.4 0.2

Not a Valid Probability Distribution

1.  a. $p\left(3\right)=0.1$, 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$\approx1$, e. $\approx0$

## Answers to Chapter 4 Exercises

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

## Answers to Chapter 5 Exercises

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