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Chapter 8: Q-8-1E (page 465)

Question: The purpose of this exercise is to compare the variability of with the variability of .

a. Suppose the first sample is selected from a population with mean and variance . Within what range should the sample mean vary about of the time in repeated samples of measurements from this distribution? That is, construct an interval extending standard deviations of on each side of .

b. Suppose the second sample is selected independently of the first from a second population with mean and variance . Within what range should the sample mean vary about the time in repeated samples of measurements from this distribution? That is, construct an interval extending standard deviations on each side .

c. Now consider the difference between the two sample means . What are the mean and standard deviation of the sampling distribution ?

d. Within what range should the difference in sample means vary about the time in repeated independent samples of measurements each from the two populations?

e. What, in general, can be said about the variability of the difference between independent sample means relative to the variability of the individual sample means?

Short Answer

Expert verified

Answer

The standard deviation is a statistic that calculates the square root of the variance as well as quantifies the dispersal of a collection compared to its average.

Step by step solution

01

Central Limit Theorem. 

According to theCentral Limit Theorem,the sampling distribution of the sample means approaches a normal distribution, irrespective of the shape of population distribution if the sample size is over 30.

02

(a) Find the interval extending 2 standard deviations x1¯on each sideμ1

d

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Most popular questions from this chapter

Estimating production time.A widely used technique for estimating the length of time it takes workers to produce a product is the time study. In a time study, the task to be studied is divided into measurable parts, and each is timed with a stopwatch or filmed for later analysis. For each worker, this process is repeated many times for each subtask. Then the average and standard deviation of the time required to complete each subtask are computed for each worker. A worker鈥檚 overall time to complete the task under study is then determined by adding his or her subtask-time averages (Gaither and Frazier, Operations Management, 2001). The data (in minutes) given in the table are the result of a time study of a production operation involving two subtasks.


Worker AWorker B

Repetition

Subtask 1

Subtask 2

Subtask 1

Subtask 2

1

30

2

31

7

2

28

4

30

2

3

31

3

32

6

4

38

3

30

5

5

25

2

29

4

6

29

4

30

1

7

30

3

31

4

a.Find the overall time it took each worker to complete the manufacturing operation under study.

b.For each worker, find the standard deviation of the seven times for subtask 1.

c.In the context of this problem, what are the standard deviations you computed in part bmeasuring?

d.Repeat part b for subtask 2.

e.If you could choose workers similar to A or workers similar to B to perform subtasks 1 and 2, which type would you assign to each subtask? Explain your decisions on the basis of your answers to parts a鈥揹.

Independent random samples from normal populations produced the results shown in the next table.

Sample 1


Sample 2

1.23.11.72.83.0

 4.22.73.63.9

a. Calculate the pooled estimate of 蟽2.

b. Do the data provide sufficient evidence to indicate that 渭2&驳迟;渭1? Test using 伪=.10.

c. Find a 90% confidence interval for (渭1鈭�渭2).

d. Which of the two inferential procedures, the test of hypothesis in part b or the confidence interval in part c, provides more information about (渭1鈭�渭2)?

Ages of self-employed immigrants. Is self-employment for immigrant workers a faster route to economic advancement in the country? This was one of the questions studied in research published in the International Journal of Manpower (Vol. 32, 2011). One aspect of the study involved comparing the ages of self-employed and wage-earning immigrants. The researcher found that in Sweden, native wage earners tend to be younger than self-employed natives. However, immigrant wage earners tend to be older than self-employed immigrants. This inference was based on the table's summary statistics for male Swedish immigrants.

Self-employed immigrants

Wage-earning immigrants

Sample Size

870

84,875

Mean

44.88

46.79

Source: Based on L. Andersson, "Occupational Choice and Returns to Self-Employment Among Immigrants," International Journal of Manpower, Vol. 32, No. 8, 2011 (Table I).

a. Based on the information given, why is it impossible to provide a measure of reliability for the inference "Self-employed immigrants are younger, on average, than wage-earning immigrants in Sweden"?

b. What information do you need to measure reliability for the inference, part a?

c. Give a value of the test statistic that would conclude that the true mean age of self-employed immigrants is less than the true mean age of wage-earning immigrants if you are willing to risk a Type I error rate of .01.

d. Assume that s, the standard deviation of the ages is the same for both self-employed and wage-earning immigrants. Give an estimate of s that would lead you to conclude that the true mean age of self-employed immigrants is less than the true mean age of wage-earning immigrants using 伪=0.01 .

e. Is the true value of s likely to be larger or smaller than the one you calculated in part d?

Given the following values of x, s, and n, form a 90% confidence interval for蟽2

a. x=21,s=2.5,n=50

b. x=1.3,s=0.02,n=15

c. x=167,s=31,n=22

d.x=9.4,s=1.5,n=5

Conducting a political poll. A pollster wants to estimate the difference between the proportions of men and women who favor a particular national candidate using a 90% confidence interval of width .04. Suppose the pollster has no prior information about the proportions. If equal numbers of men and women are to be polled, how large should the sample sizes be?
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