5.8. Dice. When two balanced dice are rolled, 36 equally likely outcomes are possible, as depicted in Fig. 4.1 on page 154. Let \(Y\) denote the sum of the dice.

1. What are the possible values of the random variable \(Y\) ?

Use random-variable notation to represent the event that the sum of the dice is 7.

Find \(\Pr \left( Y=7 \right)\).

d. Find the probability distribution of \(Y\).Leave your probabilities in fraction form.

e. Construct a probability histogram for \(Y\).

5.18. Dice. In Exercise 5.8 on page 233, you were asked to determine the probability distribution of the sum of the dice, \(Y\), when two balanced dice are rolled. That probability distribution is as follows.

1. Find and interpret the mean of \(Y\).
2. Obtain the standard deviation of \(Y\) by using one of the formulas given in Definition 5.5 on page 237.
3. Draw a probability histogram for the random variable; locate the mean; and show one, two, and three standard deviation intervals.

5.20. Evaluating Investments. An investor plans to put $50,000 in one of four investments. The return on each investment depends on whether next year's economy is strong or weak. The following table summarizes the possible payoffs, in dollars, for the four investments.

Let V, W, X, and Y denote the payoffs for the certificate of deposit, office complex, land speculation, and technical school, respectively. Then V, W, X, and Y are random variables. Assume that next year's economy has a 40% chance of being strong and a 60% chance of being weak.

1. Find the probability distribution of each random variable V, W, X, and Y
2. Determine the expected value of each random variable.
3. Which investment has the best expected payoff? Which has the worst?
4. Which investment would you select? Explain.

5.28. Evaluate the following binomial coefficients.

1. \(\left( \begin{matrix}
   4 \\
   1 \\
   \end{matrix} \right)\)
2. \(\left( \begin{matrix}
   6 \\
   2 \\
   \end{matrix} \right)\)
3. \(\left( \begin{matrix}
   8 \\
   3 \\
   \end{matrix} \right)\)
4. \(\left( \begin{matrix}

9 \\

6 \\

\end{matrix} \right)\)

5.34 For each of the following probability histograms of binomial distributions, specify whether the success probability is less than, equal to, or greater than 0.5. Explain your answers.

5.36, Recidivism. In the May 2003 issue of Scientific American, R. Doyle examined rehabilitation of felons in the article, "Reducing Crime: Rehabilitation is Making a Comeback." One aspect of the article discussed recidivism of juvenile prisoners between 14 and 17 years old, indicating that 82% of those released in 1994 were rearrested within 3 years. Assuming that recidivism rate still applies today, solve the following problems for six newly released juvenile prisoners between 14 and 17 years old.

1. Determine the probability that the number rearrested within 3 years will be exactly four; at least four; at most five; between two and five, inclusive.
   
2. Determine the probability distribution of the random variable \(Y\) , the number of released prisoners of the six who are rearrested within 3 years.
3. Determine and interpret the mean of the random variable \(Y\).
4. Obtain the standard deviation of \(Y\).
5. If, in fact, exactly two of the six released juvenile prisoners are rearrested within 3 years, would you be inclined to conclude that the recidivism rate today has decreased First consider the probability \(\Pr (Y\le 2)\). First consider the probability \(\Pr (Y\le 2)\).

6.4. Which normal distribution has a wider spread: the one with mean 1 and standard deviation 2 or the one with mean 2 and standard deviation 1? Explain your answer.

6.8. Sketch the normal distribution with

1. \(\mu =3,\ \sigma =3\)
2. \(\mu =1,\ \sigma =3\) and
3. \(\mu =3,\ \sigma =1\).

6.20. With which normal distribution is the standard normal curve associated?

6.30. Find the area under the standard normal curve that lies to the right of

1. -1.07
2. 0.6
3. 0
4. 4.2

6.32. Determine the area under the standard normal curve that lies

1. either to the left of -2.12 or to the right of 1.67.
2. either to the left of 0.63 or to the right of 1.54.

6.34. In each part, find the area under the standard normal curve that lies between the specified z-scores, sketch a standard normal curve, and shade the area of interest.

1. -1 and 1
2. -2 and 2
3. -3 and 3

6.3. Obtain the z-score for which the area under the standard normal curve to its left is 0.025.

6.38. Obtain the z-score that has an area of 0.95 to its right.

6.40. Find the following \z-scores.

1. \({{z}_{0.03}}\)
2. \({{z}_{0.005}}\)

6.42. Complete the following table.

6.48 Metastatic Carcinoid Tumors. A study of sizes of metastatic carcinoid tumors in the heart was conducted by U. Pandya et al. and published as the article, "Metastatic Carcinoid Tumor to the Heart: Echocardiographic-Pathologic Study of 11 Patients". Based on that study, we assume that lengths of metastatic carcinoid tumors in the heart are normally distributed with mean 1.8 cm and standard deviation 0.5 cm. Find the percentage of metastatic carcinoid tumors in the heart that

1. are between 1 cm and 2 cm long.
2. exceed 3 cm in length.

6.50 Polychaete Worms. Opisthotrochopodus n.sp. is a polychaete worm that inhabits deep sea hydrothermal vents along the Mid-Atlantic Ridge. According to an article by Van Dover et al. in Marine Ecology Progress Series the lengths of female polychaete worms are normally distributed with mean 6.1 mm and standard deviation 1.3 mm. Let \(X\) denote the length of a randomly selected female polychaete worm. Determine

1. \(\Pr (X\le 3)\).
2. \(\Pr (5<X<7)\)

Solution: a) We have that

b) Now we have

6.54. Metastatic Carcinoid Tumors. Refer to Exercise 6.48.

1. Determine the quartiles for lengths of metastatic carcinoid tumors in the heart.
2. Obtain the 20th percentile.
3. Find the seventh decile.
4. Interpret your answers for parts (a)-(c).

6.66. Cell Phone Rates. In the February 2003 issue of Consumer Reports, different cell phone providers and plans were compared. The monthly fees, in dollars, for a sample of the providers and plans are shown in the following table.

7.18. Although, in general, you cannot know the sampling distribution of the sample mean exactly, by what distribution can you often approximate it?

7.20. Does the sample size have an effect on the mean of all possible sample means? Explain your answer.

7.22. Explain why increasing the sample size tends to result in a smaller sampling error when a sample mean is used to estimate a population mean.

7.30. Working at Home. According to the U.S. Bureau of Labor Statistics publication News, self-employed persons with home-based businesses work a mean of 23 hours per week at home with a standard deviation of 10 hours.

1. Identify the population and variable.
2. For samples of size 100, find the mean and standard deviation of all possible sample mean hours worked per week at home.
3. Repeat part (b) for samples of size 1000 .

7.32. Menopause in Mexico. In the journal article "Age at Menopause in Puebla, Mexico" (Human Biology, Vol. 75, No. 2, pp. 205-206), authors L. Sievert and S. Hautaniemi compared the age of menopause for different populations. Menopause, the last menstrual period, is a universal phenomenon among females. According to the article, the mean age of menopause, surgical or natural, in Puebla, Mexico is 44.8 years with a standard deviation of 5.87 years. Let \(\bar{x}\) denote the mean age of menopause for a sample of females in Puebla, Mexico.

1. For samples of size 40 , find the mean and standard deviation of \(\bar{x}\). Interpret your results in words.
2. Repeat part (a) with n=120.

7.40. A variable of a population has a mean of \(\mu =35\) and a standard deviation of \(\sigma =42\).

1. If the variable is normally distributed, identify the sampling distribution of the sample mean for samples of size 9.

7.42. A variable of a population has mean \(\mu \) and standard deviation \(\sigma \). For a large sample size n, answer the following questions.

1. Identify the distribution of \(\bar{x}\).
2. Does your answer to part (a) depend on n being large? Explain your answer.
3. Identify the mean and the standard deviation of \(\bar{x}\).
4. Does your answer to part (c) depend on the sample size being large? Why or why not?

7.46. Teacher Salaries. Data on salaries in the public school system are published annually in National Survey of Salaries classroom teachers is $43,658. Assume a standard deviation of $8000.

1. Determine the sampling distribution of the sample mean for samples of size 64. Interpret your answer in terms of the distribution of all possible sample mean salaries for samples of 64 teachers.
2. Repeat part (a) for samples of size 256.
3. Do you need to assume that public school teacher salaries are normally distributed to answer parts (a) and (b)? Explain your answer.

7.50. Teacher Salaries. Refer to Exercise 7.46.

1. Determine the percentage of all samples of 64 public school teachers that have mean salaries within $1000 of the population mean salary of $43,658. Interpret your answer in terms of sampling error.
2. Repeat part (a) for samples of size 256.

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