Mastery 6

1.

A sample of 135 results in 108 successes. Use Table 1.

a.

Calculate the point estimate for the population proportion of successes. (Do not round intermediate calculations. Round your answer to 3 decimal places.)

Point estimate

b.

Construct an 99% and a 95% confidence intervals for the population proportion. (Round intermediate calculations to 4 decimal places, "z" value to 2 decimal places, and final answers to 3 decimal places.)

Confidence Level Confidence Interval

99% to

95% to

c.

Can we conclude at 99% confidence that the population proportion differs from 0.860?

No, since the confidence interval does not contain the value 0.860.

No, since the confidence interval contains the value 0.860.

Yes, since the confidence interval does not contain the value 0.860.

Yes, since the confidence interval contains the value 0.860.

d.

Can we conclude at 95% confidence that the population proportion differs from 0.860?

No, since the confidence interval contains the value 0.860.

No, since the confidence interval does not contain the value 0.860.

Yes, since the confidence interval contains the value 0.860.

Yes, since the confidence interval does not contain the value 0.860.

rev: 08_02_2013_QC_32363, 08_20_2013_QC_33737, 08_2

2.

Last year, the typical college student graduated with $27,400 in debt (The Boston Globe, May 27, 2012). Let debt among recent college graduates be normally distributed with a standard deviation of $5,000. Use Table 1.

a.

What is the probability that the average debt of three recent college graduates is more than $20,000? (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

Probability

b.

What is the probability that the average debt of three recent college graduates is more than $25,000? (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

Probability

rev: 08_01_2013_QC_33083

3.

The monthly closing stock prices (rounded to the nearest dollar) for Panera Bread Co. for the first six months of 2010 are reported in the following table. Use Table 2.

Months Closing Stock Price

January 2010 $5

February 2010 6

March 2010 9

April 2010 8

May 2010 10

June 2010 4

SOURCE: http://finance.yahoo.com .

a.

Calculate the sample mean and the sample standard deviation. (Round intermediate calculations to 4 decimal places and "sample mean" and "sample standard deviation" to 2 decimal places.)

Sample mean

Sample standard deviation

b.

Compute the 90% confidence interval for the mean stock price of Panera Bread Co., assuming that the stock price is normally distributed. (Round "t" value to 3 decimal places, and final answers to 2 decimal places.)

Confidence interval to

c.

What happens to the margin of error if a higher confidence level is used for the interval estimate?

The margin of error increases as the confidence level increases.

The margin of error decreases as the confidence level increases.

rev: 03_30_2013_QC_28735, 08_02_2013_QC_32363, 08_20_2013_QC_33737

4.

A hair salon in Cambridge, Massachusetts, reports that on seven randomly selected weekdays, the number of customers who visited the salon were 83, 46, 38, 34, 47, 42, and 48. It can be assumed that weekday customer visits follow a normal distribution. Use Table 2.

a.

Construct a 90% confidence interval for the average number of customers who visit the salon on weekdays. (Round intermediate calculations to 4 decimal places, "sample mean" and "sample standard deviation" to 2 decimal places and "t" value to 3 decimal places, and final answers to 2 decimal places.)

Confidence interval to

b.

Construct a 99% confidence interval for the average number of customers who visit the salon on weekdays. (Round intermediate calculations to 4 decimal places, "sample mean" and "sample standard deviation" to 2 decimal places and "t" value to 3 decimal places, and final answers to 2 decimal places.)

Confidence interval to

c.

What happens to the width of the interval as the confidence level increases?

As the confidence level increases, the interval becomes wider and less precise.

As the confidence level increases, the interval becomes narrower and less precise.

rev: 08_20_2013_QC_33737

5.

Acceptance sampling is an important quality control technique, where a batch of data is tested to determine if the proportion of units having a particular attribute exceeds a given percentage. Suppose that 12% of produced items are known to be nonconforming. Every week a batch of items is evaluated and the production machines are adjusted if the proportion of nonconforming items exceeds 16%. Use Table 1.

a.

What is the probability that the production machines will be adjusted if the batch consists of 56 items? (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

Probability

b.

What is the probability that the production machines will be adjusted if the batch consists of 112 items? (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

Probability

rev: 04_23_2013_QC_29568, 08_01_2013_QC_33083

6.

Given the recent economic downturn, only 63% in a graduating class of 263 will find employment in the first round of a job search. You have 21 friends who have recently graduated. Assume that your friends represent a random sample. Use Table 1.

a-1.

Interpret the sampling distribution of the sample proportion of your friends who will find employment in the first round of a job search.

The sampling distribution is not approximately normal and we need to apply the finite population correction.

The sampling distribution is approximately normal and we need to apply the finite population correction.

The sampling distribution is approximately normal and we need not apply the finite population correction.

The sampling distribution is not approximately normal and we need not apply the finite population correction.

a-2.

Calculate the expected value and the standard error of the sample proportion. (Round intermediate calculations to 4 decimal places, "expected value" to 2 decimal places and "standard deviation" to 4 decimal places.)

Expected value

Standard error

b.

What is the probability that less than 53% of your friends will find employment in the first round of a job search? (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

Probability

rev: 04_23_2013_QC_29569, 08_01_2013_QC_3

7.

A car manufacturer is concerned about poor customer satisfaction at one of its dealerships. The management decides to evaluate the satisfaction surveys of its next 66 customers. The dealer will be fined if the number of customers who report favorably is between 26 and 33. The dealership will be dissolved if fewer than 26 report favorably. It is known that 62% of the dealer’s customers report favorably on satisfaction surveys. Use Table 1.

a.

What is the probability that the dealer will be fined? (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

Probability

b.

What is the probability that the dealership will be dissolved? (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

Probability

rev: 04_10_2013_QC_28993, 08_01_2013_QC_33083, 10_24_2013_QC_37832, 03_28_2014_QC_47367, 10_29_2014_QC_57146

8.

In order to estimate the mean 30-year fixed mortgage rate for a home loan in the United States, a random sample of 12 recent loans is taken. The average calculated from this sample is 5.30%. It can be assumed that 30-year fixed mortgage rates are normally distributed with a standard deviation of 0.3%. Compute 95% and 99% confidence intervals for the population mean 30-year fixed mortgage rate. Use Table 1. (Round intermediate calculations to 4 decimal places, "z" value and final answers to 2 decimal places. Enter your answers as percentages, not decimals.)

Confidence Level Confidence Interval

95% % to %

99% % to %

rev: 08_02_2013_QC_32363, 08_20_2013_QC_33737, 10_19_2013_QC_378

9.

At a new exhibit in the Museum of Science, people are asked to choose between 71 or 176 random draws from a machine. The machine is known to have 88 green balls and 82 red balls. After each draw, the color of the ball is noted and the ball is put back for the next draw. You win a prize if more than 59% of the draws result in a green ball. Use Table 1.

a.

Calculate the probability of getting more than 59% green balls.(Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

n Probability

71

176

b. Would you choose 71 or 176 draws for the game?

71 balls

176 balls

rev: 08_01_2013_QC_33083, 07_12_2014_QC_51430

10.

A machine that is programmed to package 2.65 pounds of cereal is being tested for its accuracy. In a sample of 49 cereal boxes, the sample mean filling weight is calculated as 2.65 pounds. It can be assumed that filling weights are normally distributed with a population standard deviation of 0.06 pound. Use Table 1.

a-1.

Identify the relevant parameter of interest for these quantitative data.

The parameter of interest is the proportion filling weight of all cereal packages.

The parameter of interest is the average filling weight of all cereal packages.

a-2.

Compute the point estimate as well as the margin of error with 99% confidence. (Round intermediate calculations to 4 decimal places. Round "z" value and final answers to 2 decimal places.)

Point estimate

Margin of error

b-1.

Calculate the 99% confidence interval. (Use rounded margin of error. Round your answers to 2 decimal places.)

Confidence interval to

b-2.

Can we conclude that the packaging machine is operating improperly?

Yes, since the confidence interval contains the target filling weight of 2.65.

Yes, since the confidence interval does not contain the target filling weight of 2.65.

No, since the confidence interval contains the target filling weight of 2.65.

No, since the confidence interval does not contain the target filling weight of 2.65.

c.

How large a sample must we take if we want the margin of error to be at most 0.01 pound with 99% confidence? (Round intermediate calculations to 4 decimal places. Round "z" value to 2 decimal places and round up your final answer to the next whole number.)

Sample size

rev: 03_29_2013_QC_28850, 08_02_2013_QC_32363, 08_20_2013_QC_33737, 11_19_2013_QC_39592