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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

Independent random samples from two populations with standard deviations σ1=2andσ2=8, respectively, are selected. The sample sizes and the sample means are recorded in the following table:

Sample 1

Sample 2

n1=58x¯1=17.5

n2=62x¯2=16.23

a. Calculate the standard error of the sampling distribution for Sample 1.

b. Calculate the standard error of the sampling distribution for Sample 2.

c. Suppose you were to calculate the difference between the sample means (x1−x2). Find the mean and standard error of the sampling distribution (x1−x2).

d. Will the statistic (x1−x2) be normally distributed?

Traffic sign maintenance. Refer to the Journal of Transportation Engineering (June 2013) study of traffic sign maintenance in North Carolina, Exercise 8.54 (p. 489). Recall that the proportion of signs on NCDOT-maintained roads that fail minimum requirements was compared to the corresponding proportion for signs on county-owned roads. How many signs should be sampled from each maintainer to estimate the difference between the proportions to within .03 using a 90% confidence interval? Assume the same number of signs will be sampled from NCDOT-maintained roads and county-owned roads

Solar energy generation along highways. Refer to the International Journal of Energy and Environmental Engineering (December 2013) study of solar energy generation along highways, Exercise 8.39 (p. 481). Recall that the researchers compared the mean monthly amount of solar energy generated by east-west– and north-south– oriented solar panels using a matched-pairs experiment. However, a small sample of only five months was used for the analysis. How many more months would need to be selected to estimate the difference in means to within 25 kilowatt-hours with a 90% confidence interval? Use the information provided in the SOLAR file to find an estimate of the standard error required to carry out the calculation

Vulnerability of counting party Web spots. When you subscribe to your Facebook account, you're granted access to further. Then 1 million counting parties (RP) Web spots. Vulnerabilities in this sign-on system may permit a bushwhacker to gain unauthorized access to your profile, allowing the bushwhacker to impersonate you on the RP Web point. Computer and systems Masterminds delved into the vulnerability of counting party Web spots and presented their results at the Proceedings of the 5th AMC Factory on Computers & Communication Security (October 2012). RP Web spots were distributed as Garçon- inflow or customer- inflow Web spots. Of the 40 garçon- inflow spots studied, 20 were planted to be vulnerable to impersonation attacks. Of the 54 customer-inflow spots examined, 41 were. Plant to be vulnerable to impersonation attacks. Give your opinion on whether a customer- inflow Web point is more likely to be vulnerable to an impersonation attack than a garçon- inflow Website. However, how much more likely? If so.

Question: Two independent random samples have been selected—100 observations from population 1 and 100 from population 2. Sample means x¯1=26.6,x¯2= 15.5 were obtained. From previous experience with these populations, it is known that the variances areσ12=9andσ22=16 .

a. Find σ(x¯1-x¯2).

b. Sketch the approximate sampling distribution for (x¯1-x¯2), assuming (μ1-μ2)=10.

c. Locate the observed value of (x¯1-x¯2)the graph you drew in part

b. Does it appear that this value contradicts the null hypothesis H0:(μ1-μ2)=10?

d. Use the z-table to determine the rejection region for the test againstH0:(μ1-μ2)≠10. Useα=0.5.

e. Conduct the hypothesis test of part d and interpret your result.

f. Construct a confidence interval for μ1-μ2. Interpret the interval.

g. Which inference provides more information about the value of μ1-μ2— the test of hypothesis in part e or the confidence interval in part f?
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