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A survey was designed to study how business operations vary according to their size. Companies were classified as small, medium, or large. Questionnaires were sent to 200 randomly selected businesses of each size. Because not all questionnaires were returned, researchers decided to investigate the relationship between the response rate and the size of the business. The data are given in the two-way table.
\(\text{Business size} \\ \begin{array}{l|r|r|r|r} && \text { Small } & \text { Medium } & \text { Large } & \text { Total } \\ \hline \text{Response?} & \text { Yes } & 125 & 81 & 40 & 246 \\ \hline &\text { No } & 75 & 119 & 160 & 354 \\ \hline &\text { Total } & 200 & 200 & 200 & 600 \end{array}\)
Which variable should be used as the explanatory variable? Explain.

A survey was designed to study how business operations vary according to their size. Companies were classified as small, medium, or large. Questionnaires were sent to 200 randomly selected businesses of each size. Because not all questionnaires were returned, researchers decided to investigate the relationship between the response rate and the size of the business. The data are given in the two-way table.
\(\begin{array} {lc} & \text{Business size} \ \text {Response?} &\begin{array}{l|c|c|c|c} & \text { Small } & \text { Medium } & \text { Large } & \text { Total } \\ \hline \text { Yes } & 125 & 81 & 40 & 246 \\ \hline \text { No } & 75 & 119 & 160 & 354 \\ \hline \text { Total } & 200 & 200 & 200 & 600 \end{array} \end{array}\)
Construct a segmented bar chart to summarize the relationship between business size and whether or not the business replied to the survey.

In government data, a household consists of all occupants of a dwelling unit, while a family consists of 2 or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and family size in the United States.
\( \begin{array}{||c|c|c|c|c|c|c|} \hline \text{Number of people}& 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \begin{array}{l} \text { Household } \ \text { probability } \end{array} & 0.25 & 0.32 & 0.17 & 0.15 & 0.07 & 0.03 & 0.01 \\ \hline \begin{array}{l} \text { Family } \ \text { probability } \end{array} & 0 & 0.42 & 0.23 & 0.21 & 0.09 & 0.03 & 0.02 \\ \hline \end{array}\)
Let H = the number of people in a randomly selected U.S. household and F = the number of people in a randomly chosen U.S. family. Find the expected value of each random variable. Explain why this difference makes sense.

In government data, a house-hold consists of all occupants of a dwelling unit, while a family consists of 2 or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and family size in the United States.
\( \begin{array}{||c|c|c|c|c|c|c|} \hline \text{Number of people}& 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \begin{array}{l} \text { Household } \ \text { probability } \end{array} & 0.25 & 0.32 & 0.17 & 0.15 & 0.07 & 0.03 & 0.01 \\ \hline \begin{array}{l} \text { Family } \ \text { probability } \end{array} & 0 & 0.42 & 0.23 & 0.21 & 0.09 & 0.03 & 0.02 \\ \hline \end{array}\)
Let H = the number of people in a randomly selected U.S. household and F = the number of people in a randomly chosen U.S. family. The standard deviations of the 2 random variables are σH=1.421 and σF=1.249.. Explain why this difference makes sense.

Five people have a genetic disease and one child each. The random variable x is the number of children among the five who inherit the​ disease.
a)Find the mean.
b)Find the standard deviation.
\(\begin{array}{|c|c|} \hline x&0&1&2&3&4&5\\ \hline P(x)&0.027&0.162&0.311&0.311&0.162&0.027 \\ \hline \end{array}\)

Anystate Auto Insurance Company took a random sample of 370 insurance claims paid out during a 1-year period. The average claim paid was $1570. Assume \(\sigma\ math=$250\). Find 0.90 and 0.99 confidence intervals for the mean claim payment.

A random variable X has the discrete uniform distribution
\(f(x;k)=\frac{1}{k},x=1,2...,k\)
\(f(x;k)=0,\) elsewhere.
Show that the moment-generating function of X is
\(\displaystyle{M}_{{{x}}}{\left({t}\right)}={\frac{{{e}^{{{t}}}{\left({1}-{e}^{{{k}{t}}}\right)}}}{{{k}{\left({1}-{e}^{{{t}}}\right)}}}}\).

You randomly survey students about participating in the science fair. The two-way table shows the results. How many female students do not participate in the science fair?
\(\begin{array}{|c|c|}\hline & \text{No} & \text{Yes} \\ \hline \text{Gender} \\ \hline \text{Female} & 15 & 22 \\ \hline \text{Male} & 12 & 32 \\ \hline \end{array}\)

Compute the distribution of \(X+Y\) in the following cases:
X and Y are independent normal random variables with respective parameters \((\mu_{1},\sigma_{1}^{2}) and (\mu_{2},\sigma_{2}^{2})\).