Mathematics

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1. A simple random sample of 40 items resulted in a sample mean of 10. The population standard deviation is σ = 6. Round your answers to two decimal places.

a.  What is the standard error of the mean, σx?

b.  At 95% confidence, what is the margin of error?

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2. Business Week conducted a survey of graduates from 30 top MBA programs (Business Week, September 22, 2003). The survey found that the average annual salary for male and female graduates 10 years after graduation was $168,000 and $117,000, respectively. Assume the population standard deviation for the male graduates is $35,000, and for the female graduates it is $25,000. When calculating values for z, round to two decimal places.

a. What is the probability that a simple random sample of 40 male graduates will provide a sample mean within $10,000 of the population mean, $168,000 (to 4 decimals)?

b. What is the probability that a simple random sample of 40 female graduates will provide a sample mean within $10,000 of the population mean, $117,000 (to 4 decimals)?

c. In which of the preceding two cases, part (a) or part (b), do we have a higher probability of obtaining a sample estimate within $10,000 of the population mean?

d. What is the probability that a simple random sample of 100 male graduates will provide a sample mean more than $4,000 below the population mean (to 4 decimals)?

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3. In a survey, the planning value for the population proportion is p* = 0.25. How large a sample should be taken to provide a 95% confidence interval with a margin of error of 0.06?

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4. The College Board reported the following mean scores for the three parts of the SAT (The World Almanac, 2009):  Excel File: data07-21.xls  http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch07/7.21.gif

Assume that the population standard deviation on each part of the test is σ = 100.

a.  What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the Critical Reading part of the test? Round your answer to four decimal places.

b.  What is the probability that a random sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 515 on the Mathematics part of the test? Round your answer to four decimal places.

Compare this probability to the value computed in part (a).

c.  What is the probability that a random sample of 100 test takers will provide a sample mean test score within 10 of the population mean of 494 on the writing part of the test? Round your answer to three decimal places.

Comment on the differences between this probability and the values computed in parts (a) and (b).

The input in the box below will not be graded, but may be reviewed and considered by your instructor.

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5. The Pew Research Center Internet Project, conducted on the 25th anniversary of the Internet, involved a survey of 857 Internet users (Pew Research Center website, April 1, 2014). It provided a variety of statistics on Internet users. For instance, in 2014, 87% of American adults were Internet users. In 1995 only 14% of American adults used the Internet.

a.  The sample survey showed that 90% of respondents said the Internet has been a good thing for them personally. Develop a 95% confidence interval for the proportion of respondents who say the Internet has been a good thing for them personally (to 4 decimals).

95% Confidence Interval:  to

b.  The sample survey showed that 67% of Internet users said the Internet has generally strengthened their relationship with family and friends. Develop a 95% confidence interval for the proportion of respondents who say the Internet has strengthened their relationship with family and friends (to 4 decimals).

95% Confidence Interval:  to

c.  Fifty-six percent of Internet users have seen an online group come together to help a person or community solve a problem whereas only 25% have left an online group because of unpleasant interaction. Develop a 95% confidence interval for the proportion of Internet users who say online groups have helped solve a problem (to 4 decimals).

95% Confidence Interval:  to

d.  Compare the margin of error for the interval estimates in parts (a), (b), and (c). How is the margin of error related to sample proportion (to 2 decimals)?

The margin of error  as p gets closer to  .

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6. A simple random sample of 10 items resulted in a sample mean of 40. The population standard deviation is σ = 20.

a.  Compute the 95% confidence interval for the population mean. Round your answers to one decimal place.  Enter your answer using parentheses and a comma, in the form (n1,n2). Do not use commas in your numerical answer (i.e. use 1200 instead of 1,200, etc.)

b.  Assume that the same sample mean was obtained from a sample of 100 items. Provide a 95% confidence interval for the population mean. Round your answers to two decimal places.

c.  What is the effect of a larger sample size on the interval estimate?

Larger sample provides a  margin of error.

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7. The National Center for Education Statistics reported that 47% of college students work to pay for tuition and living expenses. Assume that a sample of 450 college students was used in the study.

a. Provide a 95% confidence interval for the population proportion of college students who work to pay for tuition and living expenses. (to 3 decimals)

Enter your answer using parentheses and a comma, in the form (n1,n2).

b. Provide a 99% confidence interval for the population proportion of college students who work to pay for tuition and living expenses. (to 3 decimals)

Enter your answer using parentheses and a comma, in the form (n1,n2).

c. What happens to the margin of error as the confidence is increased from 95% to 99%?

The margin of error becomes

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8. People end up tossing 12% of what they buy at the grocery store (Reader’s Digest, March, 2009). Assume this is the true population proportion and that you plan to take a sample survey of 540 grocery shoppers to further investigate their behavior.

Round your answers to four decimal places.

a.  Show the sampling distribution of p, the proportion of groceries thrown out by your sample respondents.

The distribution is
Mean =
http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch07/7.33.gif =

b.  What is the probability that your survey will provide a sample proportion within ±.03 of the population proportion?

c.  What is the probability that your survey will provide a sample proportion within ±.015 of the population proportion? In determining your answer, use the standard error found in part a. and the probability found using the tables in the text.

9. 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 σ = 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. Employees 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.

a.  Using the finite population correction factor, compute the standard error for each of the three firms given a sample of size 50 (to 2 decimals).

Firm A
Firm B
Firm C

b.  What is the probability that for each firm the sample mean will be within 25 of the population mean μ (to 4 decimals)?

Firm A
Firm B
Firm C

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10. Refer to the Scheer Industries example in Section 8.2. Use 6.82 days as a planning value for the population standard deviation.

a. Assuming 95% confidence, what sample size would be required to obtain a margin of error of 1 days (round up to the next whole number)?

b. Assuming 90% confidence, what sample size would be required to obtain a margin of error of 3 days (round up to the next whole number)?

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11. Health insurers are beginning to offer telemedicine services online that replace the common office visit. Wellpoint provides a video service that allows subscribers to connect with a physician online and receive prescribed treatments (Bloomberg Businessweek (March 4 – March 9, 2014). Wellpoint claims that users of its LiveHealth Online service saved a significant amount of money on a typical visit. The data shown below ($), for a sample of 20 online doctor visits, are consistent with the savings per visit reported by Wellpoint. http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch08/8-21.png

Assuming the population is roughly symmetric, construct a 95% confidence interval for the mean savings for a televisit to the doctor as opposed to an office visit (to 2 decimals).

95% confidence interval: $ to $ per visit

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12. The president of Doerman Distributors, Inc., believes that 33% of the firm’s orders come from first-time customers. A simple random sample of 100 orders will be used to estimate the proportion of first-time customers.

a.  Assume that the president is correct and p = 0.33. What is the sampling distribution of p for this study?

b.  What is the probability that the sample proportion will be between .26 and .40 (to 4 decimals)?

c.  What is the probability that the sample proportion will be between .31 and .35 (to 4 decimals)?

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13. The range for a set of data is estimated to be 68.

a.  What is the planning value for the population standard deviation?

b.  At 95% confidence, how large a sample would provide a margin of error of 4?

c.  At 95% confidence, how large a sample would provide a margin of error of 2?

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14. The following data are from a simple random sample. 4, 9, 13, 6, 13, 15

a.  What is the point estimate of the population mean?

b.  What is the point estimate of the population standard deviation (to 1 decimal)?

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15. The average cost per night of a hotel room in New York City is $266 (SmartMoney, March 2009). Assume this estimate is based on a sample of 45 hotels and that the sample standard deviation is $60.

a.  With 95% confidence, what is the margin of error? Round your answer to the nearest dollar. $

b.  What is the 95% confidence interval estimate of the population mean? Round your answers to the nearest dollar.  to

c.  Two years ago the average cost of a hotel room in New York City was $229. Discuss the change in cost over the two-year period.

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