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Chapter 6: Q3E (page 338)

Question: A random sample of n measurements was selected from a population with unknown mean渭and known standard deviation蟽2. Calculate a 95% confidence interval for伪for each of the following situations:

a. n = 75, X = 28,蟽2= 12

b. n = 200, X= 102, 蟽2= 22

c. n = 100, X= 15,蟽2=.3

d. n = 100, X= 4.05, 蟽2= .83

e. Is the assumption that the underlying population of measurements is normally distributed necessary to ensure the validity of the confidence intervals in parts a鈥揹? Explain.

Short Answer

Expert verified

Answer

a.(28.4卤27.6)b.(102.33卤101.67)c.(100.0588卤99.9412)d.(100.16268卤99.83732)e.No

Step by step solution

01

Step 1: Computing μ  when n is 75, x is 28 and σ2 is 12

a.

As the confidence interval is 95%, the significance level will be 5% which means 0.05.

Therefore,伪2=0.052

=0.025

. Now the value of z伪2can be found from the z table and so the values is 1.96.

The margin of error is calculated below:

The calculation of the confidence intervals of the boundaries are calculated below:

02

Step 1: Computing μ  when n is 200, x is 102 and σ2 is 21

b.

As the confidence interval is 95%, the significance level will be 5% which means 0.05.

Therefore,伪2=0.052

=0.025

. Now the value of can be found from the z table and so the values is 1.96.

The margin of error is calculated below:

The calculation of the confidence intervals of the boundaries are calculated below:

03

Step 1: Computing μ  when n is 100, x is 15 and σ2 is 0.3

c

As the confidence interval is 95%, the significance level will be 5% which means 0.05.

Therefore,伪2=0.052

=0.025

. Now the value of can be found from the z table and so the values is 1.96.

The margin of error is calculated below:

The calculation of the confidence intervals of the boundaries are calculated below:

04

Step 1: Computing μ  when n is 100, x is 4.05 and σ2 is 0.83

d.

As the confidence interval is 95%, the significance level will be 5% which means 0.05.

Therefore,伪2=0.052

=0.025

.

Now the value ofZ伪2can be found from the z table and so the values is 1.96.

The margin of error is calculated below:

The calculation of the confidence intervals of the boundaries are calculated below:

05

Determination of the validity of the assumption 

e.

The answer is no in this case. According to the theory of central limit theorem, the sample mean remains exactly normal. This applies specifically for the large samples where the size remains above 30.

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Most popular questions from this chapter

Unethical corporate conduct. How complicit are entrylevel accountants in carrying out an unethical request from their superiors? This was the question of interest in a study published in the journal Behavioral Research in Accounting (July 2015). A sample of 86 accounting graduate students participated in the study. After asking the subjects to perform what is clearly an unethical task (e.g., to bribe a customer), the researchers measured each subject鈥檚 intention to comply with the unethical request score. Scores ranged from -1.5 (intention to resist the unethical request) to 2.5 (intention to comply with the unethical request). Summary statistics on the 86 scores follow: x炉=2.42,s=2.84.

a. Estimate 渭, the mean intention to comply score for the population of all entry-level accountants, using a 90% confidence interval.

b. Give a practical interpretation of the interval, part a.

c. Refer to part a. What proportion of all similarly constructed confidence intervals (in repeated sampling) will contain the true value of 渭?

d. Compute the interval, x炉卤2s. How does the interpretation of this interval differ from that of the confidence interval, part a?

A random sample of 225 measurements is selected from a population, and the sample means and standard deviation are x = 32.5 and s = 30.0, respectively.

a. Use a 99% confidence interval to estimate the mean of the population, 渭.

b. How large a sample would be needed to estimate m to within .5 with 99% confidence?

c. Use a 99% confidence interval to estimate the population variance, 蟽2.

d. What is meant by the phrase99% confidence as it is used in this exercise?

Largest private companies. IPOs鈥攊nitial public offerings of stock鈥攃reate billions of dollars of new wealth for owners, managers, and employees of companies that were previously privately owned. Nevertheless, hundreds of large and thousands of small companies remain privately owned. The revenues of a random sample of 15 firms from Forbes 216 Largest Private Companies list are given in the table below

a. Describe the population from which the random sample was drawn.

b. Use a 98% confidence interval to estimate the mean revenue of the population of companies in question

c. Interpret your confidence interval in the context of the problem

d. What characteristic must the population possess to ensure the appropriateness of the estimation procedure used in part b?

e. Suppose Forbes reports that the true mean revenue of the 216 companies on the list is $5.0 billion. Is the claim believable?

Question: For the binomial sample information summarized in each part, indicate whether the sample size is large enough to use the methods of this chapter to construct a confidence interval for p.

a. n = 400,p^= .10

b. n = 50,p^= .10

c. n = 20,p^= .5

d. n = 20,p^= .3

Corporate sustainability of CPA firms. Refer to the Business and Society (March 2011) study on the sustainability behaviors of CPA corporations, Exercise 6.12 (p. 339). Recall that the level of support for corporate sustainability (measured on a quantitative scale ranging from 0 to 160 points) was obtained for each in a sample of 992 senior managers at CPA firms. The accompanying Minitab printout gives 90% confidence intervals for both the variance and standard deviation of level of support for all senior managers at CPA firms.

a. Locate the 90% confidence interval for 蟽2on the printout. Interpret the result. b. Use the sample variance on the printout to calculate the 90% confidence interval for蟽2. Does your result agree with the interval shown on the printout?

c. Locate the 90% confidence interval for on the printout.

d. Use the result, part a, to calculate the 90% confidence interval for蟽. Does your result agree with the interval shown on the printout?

e. Give a practical interpretation of the 90% confidence interval for.蟽

f. What assumption about the distribution of level of support is required for the inference, part e, to be valid? Is this assumption reasonably satisfied? (Use your answer to Exercise 4.125, p. 270.)
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