# Mathematics

Homework chapter 8 (1-15)

· Table 1 : z table

· Table 2: df table

1.

A simple random sample of 25 observations is derived from a normally distributed population with a known standard deviation of 8.2. Use Table 1.

a.

Is the condition that formula76.mml is normally distributed satisfied?

Yes

No

b.

Compute the margin of error with 80% confidence. (Round intermediate calculations to 4 decimal places, “z” value and final answer to 2 decimal places.)

Margin of error

c.

Compute the margin of error with 90% confidence. (Round intermediate calculations to 4 decimal places, “z” value and final answer to 2 decimal places.)

Margin of error

d.

Which of the two margins of error will lead to a wider interval?

The margin of error with 90% confidence.

The margin of error with 80% confidence.

2.

Consider a population with a known standard deviation of 26.8. In order to compute an interval estimate for the population mean, a sample of 64 observations is drawn. Use Table 1.

a.

Is the condition that formula80.mml is normally distributed satisfied?

Yes

No

b.

Compute the margin of error at a 95% confidence level. (Round your intermediate calculations to 4 decimal places. Round “z” value and final answer to 2 decimal places.)

Margin of error

c.

Compute the margin of error at a 95% confidence level based on a larger sample of 225 observations.(Round your intermediate calculations to 4 decimal places. Round “z” value and final answer to 2 decimal places.)

Margin of error

d.

Which of the two margins of error will lead to a wider confidence interval?

95% confidence with n = 64.

95% confidence with n = 225.

3.

In order to estimate the mean 30-year fixed mortgage rate for a home loan in the United States, a random sample of 28 recent loans is taken. The average calculated from this sample is 5.25%. It can be assumed that 30-year fixed mortgage rates are normally distributed with a standard deviation of 0.50%. Compute 90% 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

90%

%

to

%

99%

%

to

%

4.

A family is relocating from St. Louis, Missouri, to California. Due to an increasing inventory of houses in St. Louis, it is taking longer than before to sell a house. The wife is concerned and wants to know when it is optimal to put their house on the market. They ask their realtor friend for help and she informs them that the last 26 houses that sold in their neighborhood took an average time of 218 days to sell. The realtor also tells them that based on her prior experience, the population standard deviation is 72 days. Use Table 1.

a.

What assumption regarding the population is necessary for making an interval estimate for the population mean?

Assume that the central limit theorem applies.

Assume that the population has a normal distribution.

b.

Construct a 90% confidence interval for the mean sale time for all homes in the neighborhood. (Round intermediate calculations to 4 decimal places, “z” value and final answers to 2 decimal places.)

Confidence interval

to

5.

We use the t distribution for the statistical inference of the population mean when the underlying population standard deviation is not known. Under the assumption that the population is normally distributed, find tα/2,df for the following scenarios. Use Table 2. (Round your answers to 3 decimal places.)

tα/2,df

a. A 90% confidence level and a sample of 28 observations.

b. A 95% confidence level and a sample of 28 observations.

c. A 90% confidence level and a sample of 15 observations.

d. A 95% confidence level and a sample of 15 observations.

6.

Let the following sample of 8 observations be drawn from a normal population with unknown mean and standard deviation: 22, 18, 14, 25, 17, 28, 15, 21. Use Table 2.

a.

Calculate the sample mean and the sample standard deviation. (Round intermediate calculations to 4 decimal places, “sample mean” to 3 decimal places and “sample standard deviation” to 2 decimal places.)

Sample mean

Sample standard deviation

b.

Construct the 80% confidence interval for the population mean. (Round “t” value to 3 decimal places, and final answers to 2 decimal places.)

Confidence interval

to

c.

Construct the 90% confidence interval for the population mean. (Round “t” value to 3 decimal places, and final answers to 2 decimal places.)

Confidence interval

to

d.

What happens to the margin of error as the confidence level increases from 80% to 90%?

As the confidence level increases, the interval becomes wider.

As the confidence level increases, the interval becomes narrower.

7.

The Chartered Financial Analyst (CFA®) designation is fast becoming a requirement for serious investment professionals. Although it requires a successful completion of three levels of grueling exams, it also entails promising careers with lucrative salaries. A student of finance is curious about the average salary of a CFA® charterholder. He takes a random sample of 36 recent charterholders and computes a mean salary of \$158,000 with a standard deviation of \$36,000. Assuming a normal distribution, use this sampleinformation to determine the 95% confidence interval for the average salary of a CFA charterholder.Use Table 2. (Round intermediate calculations to 4 decimal places, “t” value to 3 decimal places, and final answers to the nearest whole number.)

Confidence interval

to

8.

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

\$71

February 2010

73

March 2010

76

April 2010

78

May 2010

81

June 2010

75

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.

9.

A random sample of 80 observations results in 50 successes. Use Table 1.

a.

Construct a 95% confidence interval for the population proportion of successes. (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answers to 3 decimal places.)

Confidence interval

to

b.

Construct a 95% confidence interval for the population proportion of failures. (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answers to 3 decimal places.)

Confidence interval

to

10.

In a recent poll of 760 homeowners in the United States, one in five homeowners reports having a home equity loan that he or she is currently paying off. Using a confidence coefficient of 0.90, derive an interval estimate for the proportion of all homeowners in the United States that hold a home equity loan. Use Table 1. (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answers to 3 decimal places.)

Confidence interval

to

11.

An accounting professor is notorious for being stingy in giving out good letter grades. In a large section of 140 students in the fall semester, she gave out only 5% As, 23% Bs, 42% Cs, and 30% Ds and Fs. Assuming that this was a representative class, compute a 95% confidence interval of the probability of getting at least a B from this professor. Use Table 1. (Round intermediate calculations to 4 decimal places, “z” value to 2 decimal places, and final answers to 3 decimal places.)

Confidence interval

to

12.

You need to compute a 99% confidence interval for the population mean. How large a sample should you draw to ensure that the sample mean does not deviate from the population mean by more than 1.2? (Use 6.0 as an estimate of the population standard deviation from prior studies.) Use Table 1. (Round intermediate calculations to 4 decimal places and “z” value to 2 decimal places. Round up your answer to the nearest whole number.)

Sample size

13.

In the planning stage, a sample proportion is estimated as formula204.mml = 40/50 = 0.80. Use this information to compute the minimum sample size n required to estimate p with 99% confidence if the desired margin of error E = 0.12. What happens to n if you decide to estimate p with 90% confidence? Use Table 1. (Round intermediate calculations to 4 decimal places and “z” value to 2 decimal places. Round up your answers to the nearest whole number.)

Confidence Level

n

99%

90%

14.

An analyst from an energy research institute in California wishes to precisely estimate a 99% confidence interval for the average price of unleaded gasoline in the state. In particular, she does not want the sample mean to deviate from the population mean by more than \$0.06. What is the minimum number of gas stations that she should include in her sample if she uses the standard deviation estimate of \$0.32, as reported in the popular press? Use Table 1. (Round intermediate calculations to 4 decimal places and “z” value to 2 decimal places. Round up your answer to the nearest whole number.)

Minimum number of gas stations

15.

A student of business is interested in estimating a 99% confidence interval for the proportion of students who bring laptops to campus. He wishes a precise estimate and is willing to draw a large sample that will keep the sample proportion within five percentage points of the population proportion. What is the minimum sample size required by this student, given that no prior estimate of the population proportion is available?Use Table 1. (Round intermediate calculations to 4 decimal places and “z” value to 2 decimal places. Round up your answer to the nearest whole number.)

Minimum sample size

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