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Chapter 6: Q 27E (page 348)

The following random sample was selected from a normal distribution: 4, 6, 3, 5, 9, and 3.

  1. Construct a 90% confidence interval for the population mean
  2. Construct a 95% confidence interval for the population mean
  3. Construct a 99% confidence interval for the population mean
  4. Assume that the sample means x and sample standard deviation s remain the same as those you just calculated but are based on a sample of n = 25 observations rather than n = 6 observations. Repeat parts a–c. What is the effect of increasing the sample size on the width of the confidence intervals?

Short Answer

Expert verified

A confidence interval is described as the set of numbers seen in our collection for which we anticipate discovering the number that best represents the entire population.

Step by step solution

01

(a) The calculation is given below

From the given data:

n=6χ¯=5s=2.28

The calculation is given below:

The confidence interval =5-2.01×2.286,5+2.01×2.286=3.12,6.88

02

(b) The calculation is given below

The calculation is given below:

t5,0.052=2.57

The confidence interval =5-2.57×2.286,5+2.57×2.286=2.61,7.39

03

(c) The calculation is given below

The calculation is given below:

t5,0.012=4.03

The calculation is given below:

The confidence interval =5-4.03×2.286,5+4.03×2.286=1.25,8.75

04

(d) The calculation is given below

Now, n = 25

The calculation is given below:

90% confidence interval:

CI=5±(2.015)(2.082)/25=4.161and5.839

95% confidence interval:

CI=5±(2.015)(2.082)/25=3.930and6.070

99% confidence interval:

CI=5±(2.015)(2.082)/25=3.322and6.678

As a consequence of the aforesaid results, a larger sample size has resulted in a decline in the interval on its equivalent level.

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Most popular questions from this chapter

Employees with substance abuse problems. According to the New Jersey Governor’s Council for a Drug-Free Workplace Report, 50 of the 72 sampled businesses that are members of the council admitted that they had employees with substance abuse problems. At the time of the survey, 251 New Jersey businesses were members of the Governor’s Council. Use the finite population correction factor to find a 95% confidence interval for the proportion of all New Jersey Governor’s Council business members who have employees with substance abuse problems. Interpret the resulting interval.

Use Table III, Appendix D to determine thet0 values foreach of the following probability statements and their respectivedegrees of freedom (df ).

a.Pt⩽t0=.25withdf=15

b.Pt⩾t0=.1withdf=8

c.P-t0⩽t⩽t0=.01withdf=19

d.P-t0⩽t⩽t0=.05withdf=24

“Out of control” production processes. When companies employ control charts to monitor the quality of their products, a series of small samples is typically used to determine if the process is “in control” during the period of time in which each sample is selected. (We cover quality-control charts in Chapter 13.) Suppose a concrete-block manufacturer samples nine blocks per hour and tests the breaking strength of each. During 1 hour’s test, the mean and standard deviation are 985.6 pounds per square inch (psi) and 22.9 psi, respectively. The process is to be considered “out of control” if the true mean strength differs from 1,000 psi. The manufacturer wants to be reasonably certain that the process is really out of control before shutting down the process and trying to determine the problem. What is your recommendation?

What is the confidence level of each of the following confidence intervals μ?

  1. χ¯±1.96σn
  2. χ¯±1.645σn
  3. χ¯±2,575σn
  4. χ¯±1.282σn
  5. χ¯±.99σn

USGA golf ball tests. The United States Golf Association (USGA) tests all new brands of golf balls to ensure that they meet USGA specifications. One test conducted is intended to measure the average distance traveled when the ball is hit by a machine called "Iron Byron," a name inspired by the swing of the famous golfer Byron Nelson. Suppose the USGA wishes to estimate the mean distance for a new brand to within 1 yard with 90% confidence. Assume that

past tests have indicated that the standard deviation of the distances Iron Byron hits golf balls is approximately 10 yards. How many golf balls should be hit by Iron Byron to achieve the desired accuracy in estimating the mean?
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