A tea.in of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, Mar. 2013). An example of such a network is a system for producing integrated circuit (IC) cards with two production lines set up in sequence. Items (IC cards) first pass through Line 1 and then are processed by Line 2. Toe probability distribution of the maximum capacity level of each line is shown in the next column. Assume the lines operate independently.

$$\begin{matrix} \text{Line} & \text{Maximum Capacity, x} & \text{p(x)}\\ \text{1} & \text{0} & \text{.01}\\ \text{1} & \text{12} & \text{.02}\\ \text{1} & \text{24} & \text{.02}\\ \text{1} & \text{36} & \text{.95}\\ \text{2} & \text{0} & \text{.002}\\ \text{2} & \text{35} & \text{.002}\\ \text{2} & \text{70} & \text{.996}\\ \end{matrix}$$

Now consider the network of two production lines. What is the probability that a maximum capacity level of 30 items is maintained throughout the network?

A team of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, March 2013). An example of such a network is a system for producing integrated circuit (IC) cards with two production lines set up in sequence. Items (IC cards) first pass through Line 1 , then are processed by Line 2. The probability distribution of the maximum capacity level $Y$ of each line is shown in the next table. Assume the lines operate independently.

a. Verify that the properties of discrete probability distributions are satisfied for each line in the system.

b. Find the probability that the maximum capacity level for Line 1 will exceed 30 items.

c. Repeat part $b$ for Line 2 .

d. Now consider the network of two production lines. What is the probability that a maximum capacity level of 30 items is maintained throughout the network?

$$\begin{array}{ccc} \hline \text { Line } & \text { Maximum Capacity, } y & p(y) \\ 1 & 0 & .01 \\ & 12 & .02 \\ & 24 & .02 \\ & 36 & .95 \\ 2 & 0 & .002 \\ & 35 & .002 \\ & 70 & .996 \\ \hline \end{array}$$

Josh is graduating at the end of the academic year with a BS degree in engineering. He already has an offer with a good company for $58,000. He has learned that those who continue along a technical path in the company typically receive increases of 6 percent per year. Now that Josh smells the diploma and the money, he is ready to get out and make his mark in the company. Josh’s family, however, have encouraged him to consider completing a MS degree, thereby deferring employment for 2 years. Josh has learned that in the same company, those with an MS degree who remain on a technical path earn increases of 7.2 percent. With a graduate assistantship, Josh will have to pay only$8,000 per year out of his own pocket. He will be 23 when he graduates with the BS degree, 25 when he completes an MS degree, and 65 when he retires. All salaries and costs are considered to occur at the end of the year. Based on the PW of salaries and costs at Josh’s TVOM of 7 percent, what must his minimum starting salary be to justify staying for the MS degree?

Evaporation from swimming pools. A new formula for estimating the water evaporation from occupied swimming pools was proposed and analyzed in the journal Heating/ Piping/Air Conditioning Engineering (April 2013). The key components of the new formula are number of pool occupants, area of pool's water surface, and the density difference between room air temperature and the air at the pool's surface. Data were collected from a wide range of pools for which the evaporation level was known. The new formula was applied to each pool in the sample, yielding an estimated evaporation level. The absolute value of the deviation between the actual and estimated evaporation level was then recorded as a percentage. The researchers reported the following summary statistics for absolute deviation percentage: $\bar{x}=18, s=20$. Assume that the sample contained $n=15$ swimming pools. Estimate the true mean absolute deviation percentage for the new formula with a $90 \%$ confidence interval.

Evaporation from swimming pools. A new formula for estimating the water evaporation from occupied swimming pools was proposed and analyzed in the journal Heating/ Piping/Air Conditioning Engineering (April 2013). The key components of the new formula are number of pool occupants, area of pool's water surface, and the density difference between room air temperature and the air at the pool's surface. Data were collected from a wide range of pools for which the evaporation level was known. The new formula was applied to each pool in the sample, yielding an estimated evaporation level. The absolute value of the deviation between the actual and estimated evaporation level was then recorded as a percentage. The researchers reported the following summary statistics for absolute deviation percentage: $\bar{x}=18, s=20$. Assume that the sample contained $n=15$ swimming pools. The American Society of Heating, Refrigerating, and Air-Conditioning Engineers (ASHRAE) handbook also provides a formula for estimating pool evaporation. Suppose the ASHRAE mean absolute deviation percentage is $\mu=34 \%$. (This value was reported in the article.) On average, is the new formula "better" than the ASHRAE formula? Explain.

Reliability of a manufacturing network. A team of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, March 2013). An example of such a network is a system for producing integrated circuit (IC) cards with two production lines set up in sequence. Items (IC cards) first pass through Line 1 , then are processed by Line 2. The probability distribution of the maximum capacity level $(x)$ of each line is shown below. Assume the lines operate independently. Find the mean maximum capacity for each line. Interpret the results practically.

Working on summer vacation. Is summer vacation a break from work? Not according to a Harris Interactive (July 2013) poll of U.S. adults. The poll found that $61 \%$ of the respondents work during their summer vacation, $22 \%$ do not work at all while on vacation, and $17 \%$ were unemployed. Assuming these percentages apply to the population of U.S. adults, consider the work status during summer vacation of a randomly selected adult. What is the probability that the adult works while on summer vacation?

A dust mite allergen level that exceeds 2 micrograms per gram $(\mu \mathrm{g} / \mathrm{g})$ of dust has been associated with the development of allergies. Consider a random sample of four homes, and let x be the number of homes with a dust mite level that exceeds 2 $\mu \mathrm{g} / \mathrm{g}$. The probability distribution for x, based on a study by the National Institute of Environmental Health Sciences, is shown in the following table:

$$\begin{matrix} \text{x} & \text{0} & \text{1} & \text{2} & \text{3} & \text{4}\\ \text{p(x)} & \text{.09} & \text{.30} & \text{.37} & \text{.20} & \text{.04}\\ \end{matrix}$$

a. Verify that the probabilities for x in the table sum to 1. b. Find the probability that three or four of the homes in the sample have a dust mite level that exceeds 2 $\mu \mathrm{g} / \mathrm{g}$. c. Find the probability that fewer than two homes in the sample have a dust mite level that exceeds 2 $\mu \mathrm{g} / \mathrm{g}$. d. Find E(x). Give a meaningful interpretation of the result. e. Find $\sigma$. f. Find the exact probability that x is in the interval $\mu \pm 2 \sigma$. Compare your answer with Chebyshev's rule and the empirical rule.

Repair and replacement costs of water pipes. Refer to the IHS Journal of Hydraulic Engineering (September 2012) study of water pipes, Exercise 11.12 (p. 646). Recall that a team of civil engineers used regression analysis to estimate $y$ = the ratio of repair to replacement cost of commercial pipe. The independent variable in the regression analysis was $x$ = the diameter (in millimeters) of the pipe. Data for a sample of 13 different pipe sizes are given in the table, followed by a Minitab simple linear regression printout.

a. Find the least squares line relating the ratio of repair to replacement cost ($y$) to pipe diameter ($x$) on the printout.

A die is tossed. Let $x$ be the number of spots observed on the upturned face of the die. Display the probability distribution of $x$ in graphical form.

Refer to the international Conference on Social Robotics (Vol. 6414, 2010) study of the trend in the design of social robots. Recall that in a random sample of 106 social (or service) robots designed to entertain, educate, and care for human users, 63 were built with legs only, 20 with wheels only, 8 with both legs and wheels, and 15 with neither legs nor wheels. One of the 106 social robots is randomly selected, and the design (e.g., wheels only) is noted. List the sample points for this study.

According to the Earth Policy Institute (July 2013), 60% of the world's solar energy cells are manufactured in China. Consider a random sample of 5 solar energy cells, and let x represent the number in the sample that are manufactured in China. We show that the probability distribution for x is given by the formula $p(x)=\frac{(5 !)(.6)^{x}(.4)^{5-x}}{(x !)(5-x) !}$, where $n! = (n)(n-1)(n-2) ... (2) (1)$. a. Explain why x is a discrete random variable. b. Find $p(x)$ for $x = 0, 1, 2, 3, 4,$ and 5. c. Show that the properties for a discrete probability distribution are satisfied. d. Find the probability that at least 4 of the 5 solar energy cells in the sample are manufactured in China.

Radon exposure in Egyptian tombs. Many ancient Egyptian tombs were cut from limestone rock that contained uranium. Since most tombs are not well ventilated, guards, tour guides, and visitors may be exposed to deadly radon gas. In Radiation Protection Dosimetry (December 2010), a study of radon exposure in tombs in the Valley of Kings, Luxor, Egypt (recently opened for public tours), was conducted. The radon levels-measured in becquerels per cubic meter $\left(\mathrm{Bq} / \mathrm{m}^3\right)$-in the inner chambers of a sample of 12 tombs were determined. Summary statistics follow: $\bar{x}=3,643 \mathrm{~Bq} / \mathrm{m}^3$ and $s=4,487 \mathrm{~Bq} / \mathrm{m}^3$. Use this information to estimate, with $95 \%$ confidence, the true mean level of radon exposure in tombs in the Valley of Kings. Interpret the resulting interval.

Credit/debit card market shares. The following table reports the U.S. credit/debit card industry's market share data for 2015. A random sample of $100 \mathrm{credit/debit}$ card users is to be questioned regarding their satisfaction with their card company. For simplification, assume that each card user carries just one card and that the market share percentages are the percentages of all card customers that carry each brand. What is the approximate probability that half or more of the sample of card users carry Visa? American Express?