The latest available data showed health expenditures were $\$ 8086$ per person in the United States or $17.6 \%$ of gross domestic product. Use $\$ 8086$ as the population mean and suppose a survey research firm will take a sample of 100 people to investigate the nature of their health expenditures. Suppose the population standard deviation is $\$ 2500$.

a. Show the sampling distribution of the mean amount of healthcare expenditures for a sample of 100 people.

A business executive, transferred from Chicago to Atlanta, needs to sell her house in Chicago quickly. The executive’s employer has offered to buy the house for $210,000, but the offer expires at the end of the week. The executive does not currently have a better offer but can afford to leave the house on the market for another month. From conversations with her realtor, the executive believes the price she will get by leaving the house on the market for another month is uniformly distributed between$200,000 and $225,000.

If she leaves it on the market for another month, what is the probability she will get at least$215,000 for the house?

A business executive, transferred from Chicago to Atlanta, needs to sell her house in Chicago quickly. The executive’s employer has offered to buy the house for $210,000, but the offer expires at the end of the week. The executive does not currently have a better offer but can afford to leave the house on the market for another month. From conversations with her realtor, the executive believes the price she will get by leaving the house on the market for another month is uniformly distributed between$200,000 and $225,000.

Should the executive leave the house on the market for another month? Why or why not?

A sample survey of $54$ discount brokers showed that the mean price charged for a trade of $100$ shares at $50$ per share was $\$33.77$ ($AAII ~Journal$, February $2006$). The survey is conducted annually. With the historical data available, assume a known population standard deviation of $\$15.$

Develop a $95\%$ confidence interval for the mean price charged by discount brokers for a trade of $100$ shares at $\$50$ per share.

Jack Lawler, a financial analyst, wishes to write an article about the American Association of Individual Investors' Shadow Stock portfolio (AAII). The DATAfile ShadowStocks contains a list of the 30 companies in the Shadow Stock portfolio as of March 2014. (AAII website March 27, 2014). Jack would like to choose a simple random sample of five of these businesses for an interview about management practises. a. The Shadow Stock companies are listed in column A of an Excel worksheet in the DATAfile. For each of the companies, we generated a random number in column $\mathrm{B}$. Using these random numbers, choose a simple random sample of five of these companies for Jack. b. Generate a new set of random numbers and use them to select a new simple random sample. Did you select the same companies?

A sample survey of $54$ discount brokers showed that the mean price charged for a trade of $100$ shares at $50$ per share was $\$33.77$ ($AAII ~Journal$, February $2006$). The survey is conducted annually. With the historical data available, assume a known population standard deviation of $\$15.$

Using the sample data, what is the margin of error associated with a $95\%$ confidence interval?

A researcher reports survey results by stating that the standard error of the mean is $20.$ The population standard deviation is $500.$

How large was the sample used in this survey?

Lori Jeffrey is a successful sales representative for a major publisher of college textbooks. Historically, Lori obtains a book adoption on $25\%$ of her sales calls. Viewing her sales calls for one month as a sample of all possible sales calls, assume that a statistical analysis of the data yields a standard error of the proportion of $.0625.$

Using the sampling distribution of $\bar{p}$, compute the probability that Lori will obtain book adoptions on $30\%$or more of her sales calls during a one-month period.

Lori Jeffrey is a successful sales representative for a major publisher of college textbooks. Historically, Lori obtains a book adoption on $25\%$ of her sales calls. Viewing her sales calls for one month as a sample of all possible sales calls, assume that a statistical analysis of the data yields a standard error of the proportion of $.0625.$

Let $\bar{p}$ indicate the sample proportion of book adoptions obtained during the month. Show the sampling distribution of $\bar{p}$.

Lori Jeffrey is a successful sales representative for a major publisher of college textbooks. Historically, Lori obtains a book adoption on $25\%$ of her sales calls. Viewing her sales calls for one month as a sample of all possible sales calls, assume that a statistical analysis of the data yields a standard error of the proportion of $.0625.$

How large was the sample used in this analysis? That is, how many sales calls did Lori make during the month?

Australians and Smoking. Fifteen percent of Australians smoke. Reuters reports that by introducing tough laws banning brand labels on cigarette packages, Australia hopes to reduce the percentage of people smoking to $10 \%$ by 2018 . Answer the following questions based on a sample of 240 Australians.

b. What is the probability the sample proportion will be within $\pm .04$ of the population proportion?

Fifteen percent of Australians smoke. By introducing tough laws banning brand labels on cigarette packages, Australia hopes to reduce the percentage of people smoking to 10% by 2018 (Reuters website, October 23, 2012). Answer the following questions based on a sample of 240 Australians.

a. Show the sampling distribution of $\bar{p}$, the proportion of Australians who are smokers.

b. What is the probability the sample proportion will be within \pm .04 of the population proportion?

c. What is the probability the sample proportion will be within \pm .02 of the population proportion?

A simple random sample of size $100$ is selected from a population with $p=.40.$

Show the sampling distribution of $\bar{p}$ .

Three firms carry inventories that differ in size. Firm A’s inventory contains $2000$ items, firm B’s inventory contains $5000$ items, and firm C’s inventory contains $10,000$ items. The population standard deviation for the cost of the items in each firm’s inventory is $\sigma = 144$. A statistical consultant recommends that each firm take a sample of $50$ items from its inventory to provide statistically valid estimates of the average cost per item. Managers of the small firm state that because it has the smallest population, it should be able to make the estimate from a much smaller sample than that required by the larger firms. However, the consultant states that to obtain the same standard error and thus the same precision in the sample results, all firms should use the same sample size regardless of population size.

What is the probability that for each firm the sample mean $\bar{x}$ will be within $\pm 25$ of the population mean $\mu$?