Assignment 1.docx

ASSIGNMENT I Part A 1) Sampling distributions describe the distribution of A) parameters. B) statistics. C) both parameters and statistics. D) neither parameters nor statistics. 2) The standard error of the mean A) is never larger than the standard deviation of the population. B) decreases as the sample size increases. C) measures the variability of the mean from sample to sample. D) All of the above. 3) The Central Limit Theorem is important in statistics because A) for a large n, it says the population is approximately normal. B) for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size. C) for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population. D) for any sized sample, it says the sampling distribution of the sample mean is approximately normal. 4) If the expected value of a sample statistic is equal to the parameter it is estimating, then we call that sample statistic A) unbiased. B) minimum variance. C) biased. D) random. 5) Which of the following is true about the sampling distribution of the sample mean? A) The mean of the sampling distribution is always μ. B) The standard deviation of the sampling distribution is always σ. C) The shape of the sampling distribution is always approximately normal. D) All of the above are true. 6) Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in a recent year. Suppose a sample of 100 major league players was taken. Find the approximate probability that the mean salary of the 100 players exceeded $3.5 million. A) approximately 0 B) 0.0228 C) 0.9772 D) approximately 1 7) The owner of a fish market has an assistant who has determined that the weights of catfish are normally distributed, with a mean of 3.2 pounds and a standard deviation of 0.8 pound. What percentage of samples of 4 fish will have sample means between 3.0 and 4.0 pounds? A) 84%

B) 67% C) 29% D) 16% 8) The finite population correction factor is A)

.

B)

.

C)

.

D)

.

9) Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is put into a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of 8 private colleges in the United States revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95.4, and 220.0. What value will be used as the point estimate for the mean endowment of all private colleges in the United States? A) $1,447.8 B) $180.975 C) $143.042 D) $8 10) A major department store chain is interested in estimating the mean amount its credit card customers spent on their first visit to the chain's new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results: = $50.50 and S = 20. Construct a 95% confidence interval for the mean amount its credit card customers spent on their first visit to the chain's new store in the mall assuming that the amount spent follows a normal distribution. A) $50.50 ± $9.09 B) $50.50 ± $10.12 C) $50.50 ± $11.00 D) $50.50 ± $11.08 11) Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is put into a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of 8 private colleges in the United States revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95.4, and 220.0. Summary statistics yield = 180.975 and S = 143.042. Calculate a 95% confidence interval for the mean endowment of all the private colleges in the United States assuming a normal distribution for the endowments. A) $180.975 ± $94.066 B) $180.975 ± $99.123 C) $180.975 ± $116.621 D) $180.975 ± $119.586

12) The amount of material used in making a custom sail for a sailboat is normally distributed with a standard deviation of 64 square feet. For a random sample of 15 sails, the mean amount of material used is 912 square feet. Which of the following represents a 99% confidence interval for the population mean amount of material used in a custom sail? A) 912 ± 49.2 B) 912 ± 42.6 C) 912 ± 44.3 D) 912 ± 46.8 13) In a survey of 472 personnel directors, 63% thought that they would be hiring new personnel over the next three months. Which of the following represents a 98% confidence interval for the proportion of all personnel directors planning to hire personnel over the next three months? A) 0.63 ± 0.057 B) 0.63 ± 0.047 C) 0.63 ± 0.042 D) 0.63 ± 0.052 14) A student records the time (in minutes) it takes to commute to school for seven days. Those results are: 21, 15, 13, 16, 10, 13, and 18. Assuming the population is normally distributed, develop a 95% confidence interval for the population mean. 15) You are told that a 95% confidence interval for the population mean is 17.3 to 24.5. If the population standard deviation is 18.2, how large was the sample? 16) A mother who is interested in the true proportion of R-rated movies shown on pay TV by a cable system randomly selects 98 listings and finds 14 of them are R-rated movies. In her report to the subcommittee she wants to be 98% confident that the true proportion will be in an interval which she states. She has asked you to assist her by preparing a 98% confidence interval based on the data she collected. What should she report? Use the following to answer questions 17-20. A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose that the actual capacity of a randomly selected tank has a distribution that is approximately normal with a mean of 15.0 gallons and a standard deviation of 0.15 gallons.

17) A) B)

The manufacturer of this gasoline tank considers the largest 2% of these tanks too large to put on the market. How large does a tank have to be, to be considered too large? 15 gallons. C) 15.72 gallons. 15.31 gallons. D) 16 gallons.

18). What proportion of tanks will hold between 14.75 and 15.10 gallons of gas? A) 0.35 C) 0.6997

B)

0.6563

19)

If a simple random sample of four tanks is selected, and their capacities can be considered independent, what is the probability that all four will hold between 14.75 and 15.10 gallons of gas?

A) B)

0.2397 0.6808

20)

If four tanks are randomly selected, what is the probability that their average capacity will be between 14.75 and 15.10 gallons? 0.2397 C) 0.9084 0.6808 D) 0.9962

A) B)

D)

C) D)

0.9833

0.9084 2.7988

21)

What is Standard error? Discuss its role in large sample theory

22)

What do you mean by confidence interval? How will you determine the confidence interval for normal population mean?

23)

Explain Parameter and Statistic.

24)

In a random selection of 64 of 600 road crossing in a town, the mean number of automobile accidents per year was found to 4.2 and sample standard deviation was 0.8. Construct a 95% confidence interval for the mean number of automobile accidents per crossing per year.

25)

Out of 300 households in a town 123 have LCD. Find 95% confidence limits to the true value of the proportion of the households with LCD in the whole town.

26)

A manufacturing concern wants to estimate the average amount of purchase of its product in a month by the customers whose standard error is $10. Find the sample size if the maximum error is not to exceed $3 with a probability of 0.99.

Part – B 1.

The only way to reduce both α and β simultaneously in a hypothesis test is to increase the sample size. A. TRUE B. FALSE

2.

The level of significance represents the probability of making a Type I error. A. TRUE B. FALSE

3. If the purpose of the hypothesis test is to establish that a population parameter is equal to or not equal to a specific value, the ______________________________________ A. ≤ or ≥ statement is always assigned to the alternative hypothesis B. ≠ statement is always assigned to the null hypothesis C. equality statement is always assigned to the alternative hypothesis D. equality statement is always assigned to the null hypothesis

4. A professor would like to test the hypothesis that the average number of minutes that a student needs to complete a statistics exam is equal to 45 minutes. A Type I error would occur if the professor concludes that the average exam time is _________________________________. A. equal to 45 minutes when, in reality, the average time is less than 45 minutes B. equal to 45 minutes when, in reality, the average time is not equal to 45 minutes C. not equal to 45 minutes when, in reality, the average time is equal to 45 minutes D. not equal to 45 minutes when, in reality, the average time is greater than 45 minutes

5. CoStar would like to test if the vacancy rate for "warehouse" stores is more than 8%. A Type II error would occur if CoStar concludes that the vacancy rate was ________________________. A. B. C. D.

more than 8% when, in reality, the proportion was 8% or less equal to 8% when, in reality, the proportion was 8% or less 8% or less when, in reality, the proportion was more than 8% not equal to 8% when, in reality, the proportion was equal to 8%

6.

Proportion data follow the ________ distribution, which can be approximated by the ________ distribution under the following conditions: nπ ≥ 5 and n(1 - π) ≥ 5. A. binomial, normal B. normal, exponential C. exponential, binomial D. Poisson, normal

7.

Many franchisers favor owners who operate multiple stores by providing them with preferential treatment. Suppose the Small Business Administration would like to perform a hypothesis test to investigate if 80% of franchisees own only one location using α = 0.05. A random sample of 120 franchisees found that 85 owned only one store. The critical value for this hypothesis test would be ________. A. 1.96 B. 2.33 C. 1.645 D. 2.57

8.

AT&T would like to test the hypothesis that the average revenue per retail user for Verizon Wireless customers equals $50. A random sample of 32 Verizon Wireless customers provided an average revenue of $54.70. It is believed that the population standard deviation for the revenue per retail user is $11.00. AT&T would like to set α = 0.05. Use the critical value approach to test this hypothesis. 9. Toll Brothers is a luxury home builder that would like to test the hypothesis that the average size of new homes exceeds 2,400 square feet. A random sample of 36 newly constructed homes had an average of 2,510 square feet. Assume that the standard deviation of the size for all newly constructed homes is 480 square feet. Toll Brothers would like to set α = 0.02. Use the critical value approach to test this hypothesis. 10. Hallmark would like to test the hypothesis that those celebrating Valentine's Day will spend more than an average of $125 on gifts. A random sample of 18 people celebrating Valentine's Day spent an average of $148.50 with a standard deviation of $34.90. Hallmark would like to set α = 0.01. Use the critical value approach to test this hypothesis. 11. The Department of Labor would like to test the hypothesis that the average hourly wage for recent college graduates is less than $20. A random sample of 24 recent college graduates

averaged $19.30 per hour with a standard deviation of $3.20 per hour. The Department of Labor would like to set α = 0.10. Use the critical value approach to test this hypothesis. 12. Over the past several years, the proportion of one-person households has been increasing. The Census Bureau would like to test the hypothesis that the proportion of one-person households exceeds 0.27. A random sample of 125 households found that 43 consisted of one person. The Census Bureau would like to set α = 0.05. Use the critical value approach to test this hypothesis. 13. Organizations are relying more and more on social-networking sites to research job candidates. According to CareerBuilder, 52% of organizations do not use these sites. A random sample of 150 organizations found that 67 of them do not use social-networking sites to research job candidates. Use the critical value approach to test the hypothesis that less than 52% of organizations do not use social-networking sites to research job candidates with α = 0.01. 14. Hotels.com would like to test the hypothesis that the proportion of American travelers in Europe that prefer an American-branded hotel equals 0.42. A random sample of 90 Americans found that 49 preferred American-branded hotels. Hotels.com would like to set α = 0.05. Use the critical value approach to test this hypothesis.

Part – C 1.

The following data represent a random sample of bank balances for a population of checking account customers at a large eastern bank. Based on these data, what is the 95 percent confidence interval estimate for the true population mean? $2,300 $756 $325 $1,457 $208 $2,345 $1,560 $124 $356 $3,179 $457 $789 $120 $2,760 $998 $508 $210 $789

2.

What are the disadvantages of using a small sample to estimate the population mean?

3.

One of the major oil products companies conducted a study recently to estimate the mean gallons of gasoline purchased by customers per visit to a gasoline station. To do this, a random sample of customers was selected with the following data being recorded that show the gallons of gasoline purchased. 8.7 22.4 9.5 13.3 18.9 22 14.4 35.7 19 24.9 5.7 15.7 8.9 22.5 15.9 Based on these sample data, construct and interpret a 95 percent confidence interval estimate for the population mean.

4.

A cell phone manufacturer claims that its phone will last for more than 8 hours of continuous talk time when the battery is fully charged. To test this claim a sample of n = 18 phones were tested. The results showed a sample mean of 8.2 hours and a sample standard deviation of 0.4 hour. Conduct the hypothesis test using a 0.5 level of significance and determine whether or not the company's claim is supported.

5.

Explain what is meant by a p-value.

6.

A company makes a device that can be fitted to automobile engines to improve the mileage. The company claims that if the device is installed, owners will observe a mean increase of more than

3.0 mpg. Assuming that the population standard deviation of increase is known to be 0.75 mpg, and a sample of size 64 cars is selected with an = 3.25 mpg, use the p-value approach to test the null hypothesis using a significance level of 0.05.

7.

Explain why an increase in sample size will reduce the probability of a Type II error but will not impact the probability of a Type I error.

8.

A small city is considering breaking away from the county school system and starting its own city school system. City leaders believe that more than 60 percent of residents support the idea. A poll of n = 215 residents is taken and 134 people say they support starting a city school district. Using a 0.10 level of significance, conduct a hypothesis test to determine whether this poll supports the belief of city leaders.

9.

Historically, voter turnout for political elections in Texas have been reported to be 54%. You have been assigned by a polling company to test the hypothesis that voter turnout during the most recent election was higher than 54%. You have collected a random sample of 90 registered voters from this elections and found that 54 actually voted. Use the p-value approach to test this hypothesis with α = 0.02.

10. The Department of Labor would like to test the hypothesis that the average hourly wage for recent college graduates is less than $20. A random sample of 24 recent college graduates averaged $19.30 per hour with a standard deviation of $3.20 per hour. The Department of Labor would like to set α = 0.10. Use the critical value approach to test this hypothesis. 11. The management of the Green Bay Packers football team would like to test the hypothesis that the average price of a ticket is less than $225 on the secondary market. A random sample of 40 customers paid an average of $207 for their ticket. Assume that the standard deviation of the price of tickets for Packers games is $43. The Green Bay Packers would like to set α = 0.01. Use the p-value approach to test this hypothesis. 12. AT&T would like to test the hypothesis that the average revenue per retail user for Verizon Wireless customers equals $50. A random sample of 32 Verizon Wireless customers provided an average revenue of $54.70. It is believed that the population standard deviation for the revenue per retail user is $11.00. AT&T would like to set α = 0.05. Use the p-value approach to test this hypothesis.