Sampling techniques can be used to estimate physical quantities. To estimate a large quantity, you might measure a representative small sample and find the total quantity by "scaling up."To estimate a small quantity, you might measure several of the small quantities together and "scale down." In each of the following, describe your estimation technique and answer the questions.

Example: How thick is a sheet of a paper?

Solution: One way to estimate the thickness of a sheet of paper is to measure the thickness of a ream ($500$ sheets) of paper. A particular ream was $7.5$ centimeters thick. Thus, a sheet of paper from this ream was $7.5 \mathrm{~cm} \div 500=0.015 \mathrm{~cm}$, or $0.15$ millimeter, thick.

How much does a sheet of paper weigh?How thick is a penny? a nickel? a dime? a quarter? Would you rather have your height stacked in pennies, nickels, dimes, or quarters? Explain.

Select the best answer. Suppose that 35% of the registered voters in a state are registered as Republicans, 40% as Democrats, and 25% as Independents. A newspaper wants to select a sample of 1000 registered voters to predict the outcome of the next election. If they randomly select 350 Republicans, randomly select 400 Democrats, and randomly select 250 Independents, did this sampling procedure result in a simple random sample of registered voters from this district? (a) Yes, because each registered voter had the same chance of being chosen. (b) Yes, because random chance was involved. (c) No, because not all registered voters had the same chance of being chosen. (d) No, because there were different number of registered voters selected from each party. (e) No, because not all possible groups of 1000 registered voters had the same chance of being chosen.

Scaling techniques can be used to estimate the physical quantity. To estimate a large quantity, you might measure a representative small sample and find out the total quantity by "scaling up." To estimate a small quantity, you might measure several of the small quantities together and "scale down." In each of the following, describe your estimation technique and answer the question. Example: How thick is a sheet of paper? Solution: One way to estimate the thickness of a sheet of paper is to measure the thickness of a ream ( $500$ sheets) of paper. A particular ream was $7.5$ centimeters thick. Thus, a sheet of paper from this ream was $7.5 \mathrm{~cm} \div 500=0.015 \mathrm{~cm}$, or $0.15$ millimeter, thick.

How many stars are visible in the sky on the clearest, darkest nights in your hometown?

In 1960, census results indicated that the age at which American men first married had a mean of 23.3 years. It is widely suspected that young people today are waiting longer to get married. We want to find out if the mean age of first marriage has increased during the past 40 y a) Write appropriate hypotheses. b) We plan to test our hypothesis by selecting a random sample of 40 men who married for the first time last year. Do you think the necessary assumptions for inference are satisfied? Explain. c) Describe the approximate sampling distribution model for the mean age in such samples. d) The men in our sample married at an average age of 24.2 years, with a standard deviation of 5.3 years. What’s the P-value for this result? e) Explain (in context) what this P-value means. f) What’s your conclusion?

In a large city school system with 20 elementary schools, the school board is considering the adoption of a new policy that would require elementary students to pass a test in order to be promoted to the next grade. The PTA wants to find out whether parents agree with this plan. Listed below are some of the ideas proposed for gathering data. For each, indicate what kind of sampling strategy is involved and what (if any) biases might result. a) Put a big ad in the newspaper asking people to log their opinions on the PTA website. b) Randomly select one of the elementary schools and contact every parent by phone. c) Send a survey home with every student, and ask parents to fill it out and return it the next day. d) Randomly select 20 parents from each elementary school. Send them a survey, and follow up with a phone call if they do not return the survey within a week.

A marketing consultant was hired to visit a random sample of five sporting goods stores across the state of California. Each store was part of a large franchise of sporting goods stores. The consultant taught the managers of each store better ways to advertise and display their goods. The net sales for 1 month before and 1 month after the consultant’s visit were recorded as follows for each store (in thousands of dollars):

$$\begin{matrix} \text{Store} & \text{1} & \text{2} & \text{3} & \text{4} & \text{5}\\ \text{Before visit} & \text{57.1} & \text{94.6} & \text{49.2} & \text{77.4} & \text{43.2}\\ \text{After visit} & \text{63.5} & \text{101.8} & \text{57.8} & \text{81.2} & \text{41.9}\\ \end{matrix}$$

Do the data indicate that the average net sales improved? (Use $\alpha=0.05$.) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value.

A local news agency conducted a survey about unemployment by randomly dialing phone numbers until it had gathered responses from 1000 adults in the state. In the survey, 19% of those who responded said they were not currently employed. In reality, only 6% of the adults in the state were not currently employed at the time of the survey. Which of the following best explains the difference in the two percentages? (a) The difference is due to sampling variability. We shouldn't expect the results of a random sample to match the truth about the population every time. (b) The difference is due to response bias. Adults who are employed are likely to lie and say that they are unemployed. (c) The difference is due to undercoverage. The survey included only adults and did not include teenagers who are eligible to work. (d) The difference is due to nonresponse. Adults who are employed are less likely to be available for the sample than adults who are unemployed.

Finally, you can also use the information on the Internet to value the entire corporation. This approach requires that you estimate XOM's annual free cash flows. Once you estimate the value of the entire corporation, you subtract the value of debt and preferred stock to arrive at an estimate of the company's equity value. By dividing this value by the number of shares of common stock outstanding, you calculate an alternative estimate of the stock's intrinsic value. Although this approach may take additional time and involves more judgment concerning forecasts of future free cash flows, you can use the financial statements and growth forecasts on the Internet as useful starting points.

If you go to the annual cash flow statement financials screen, you will find historical annual free cash flow values. Although these numbers are useful as a starting point to arrive at an estimate for the next year, MSN's definition of free cash flow subtracts out dividends. Therefore, to make it comparable to the measure in this text, you must add back dividends. To see MSN's definition of free cash flow (or any term), go to the bottom of your screen and click "Help." Then, under "Getting Started" you should click "A to Z Glossary." On the next screen you will see an alphabetic index; just click the first letter of the term for the definition you're interested in.

The following data were collected from a cell concentration sensor measuring absorbance in a biochemical stream. The input $x$ is the flow rate deviation (in

dimensionless units) and the sensor output $y$ is given in volts. The flow rate (input) is piecewise constant between sampling instants. The process is not at steady state initially, so $y$ can change even though $x=0$.

Fit a first-order model, $y(k)=a_1 y(k-1)+b_1 x(k-1)$, to the data using the least-squares approach. Plot the model response and the actual data. Can you also find a first-order continuous transfer function $G(s)$ to fit the data?

It is desired to control the exit temperature $T_2$ of the heat exchanger shown in Fig. mentioned by adjusting the steam flow rate $w_s$. Unmeasured disturbances occur in inlet temperature $T_1$. The dynamic behavior of the heat exchanger can be approximated by the transfer function

$$\frac{T_2^q(s)}{W_s^{\prime}(s)}=\frac{2.5}{10 s+1}[=] \frac{{ }^{\circ} \mathrm{F}}{\mathrm{lb} / \mathrm{s}}$$

where the time constant has units of seconds and the primes denote deviation variables. The control valve and temperature transmitter have negligible dynamics and steady-state gains of $K_v=0.2 \mathrm{lb} / \mathrm{s} / \mathrm{mA}$ and $K_m=0.25 \mathrm{~mA} /{ }^{\circ} \mathrm{F}$. Design a minimal prototype controller (i.e., Dahlin's controller with $\lambda=0$ ) that is physically realizable and based on a unit step change in the set point. Assume that a zero-order hold is used and that the sampling period is $\Delta t=2 \mathrm{~s}$.

The following table shows the Myers-Briggs personality preferences for a random sample of 406 people in the listed professions. E refers to extroverted and I refers to introverted.

$$\begin{array}{c}\text{Personality Preference Type} \\\begin{array}{l|c|c|c} \text { Occupation } & \text { E } & \text { I } & \text { Row Total } \\ \hline \text { Clergy (all denominations) } & 62 & 45 & 107 \\ \hline \text { M.D. } & 68 & 94 & 162 \\ \hline \text { Lawyer } & 56 & 81 & 137 \\ \hline \text { Column Total } & 186 & 220 & 406 \\ \hline \end{array} \end{array}$$

Use the chi-square test to determine if the listed occupations and personality preferences are independent at the 0.05 level of significance. Find the value of the chi-square statistic for the sample. Are all the expected frequencies greater than 5? What sampling distribution will you use? What are the degrees of freedom?

According to the American Automobile Association (AAA), the average cost of a gallon of regular unleaded fuel at gas stations in May 2014 was $3.65 (AAA Fuel Gauge Report). Assume that the standard deviation of such costs is$.15. Suppose that a random sample of n = 100 gas stations is selected from the population and the cost per gallon of regular unleaded fuel is determined for each. Consider $\bar{x}$, the sample mean cost per gallen. a. Calculate $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$. b. What is the approximate probability that the sample has a mean fuel cost between $3.65 and$3.67? c. What is the approximate probability that the sample has a mean fuel cost that exceeds $3.67? d. How would the sampling distribution of x change if the sample size n were doubled from 100 to 200? How do your answers to parts b and e change when the sample size is doubled?

Nationally, about 11% of the total U.S. wheat crop is destroyed each year by hail. An insurance company is studying wheat hail damage claims in Weld County, Colorado. A random sample of 16 claims in Weld County gave the following data (1% wheat crop lost to hail).

$$\begin{array}{rrrrrrrr} 15 & 8 & 9 & 11 & 12 & 20 & 14 & 11 \\ 7 & 10 & 24 & 20 & 13 & 9 & 12 & 5 \end{array}$$

The sample mean is $\bar{x}=12.5 \%$. Let x be a random variable that represents the percentage of wheat crop in Weld County lost to hail. Assume that x has a normal distribution and $\boldsymbol{\sigma}=5.0 \%$. Do these data indicate that the percentage of wheat crops lost to hail in Weld County is different (either way) from the national mean of 11%? Use $\alpha=0.01$. Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value.

In a large city school system with 20 elementary schools, the school board is considering the adoption of a new policy that would require elementary students to pass a test in order to be promoted to the next grade. The PTA wants to find out whether parents agree with this plan. Listed below are some of the ideas proposed for gathering data. For each, indicate what kind of sampling strategy is involved and what (if any) biases might result. a) Put a big ad in the newspaper asking people to log their opinions on the PTA Web site. b) Randomly select one of the elementary schools and contact every parent by phone. c) Send a survey home with every student, and ask parents to fill it out and return it the next day. d) Randomly select 20 parents from each elementary school. Send them a survey, and follow up with a phone call if they do not return the survey within a week.