DATA FILE 4
Assignment 4
1. What is the confidence level for each of the following confidence intervals for µ?
a.
b.
c.
d.
e.
2. A random sample of n measurements was selected from a population with unknown mean µ and standard deviation Ϭ. Calculate a 95% confidence interval for µ for each of the following situations:
a. n= 75, = 28, s2 = 12
b. n= 200, = 102, s2 = 22
c. n= 100, = 15, s = 3
d. n= 100, =4.05, s = 0.83
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 -d? Explain.
3. Named for the section of the 1978 Internal Revenue Code that authorized them, 401(k) plans permit employees to shift part of their before-tax salaries into investments such as mutual funds. One company, concerned with what it believed was a low employee participation rate in its 401(k) plan, sampled 30 other companies with similar plans and asked for their 401(k) participation rates. The following rates (in percentages) were obtained:
80 76 81 77 82 80 85 60 80 79 82 70
88 85 80 79 83 75 87 78 80 84 72 75
90 84 82 77 75 86
Descriptive statistics from SPSS for the data are as follows:
Number of Valid Observations (Listwise) = 30
Variable Mean Std Dev Minimum Maximum N Label
PARTRATE 79.73 5.96 60.00 90.00 30
a. Use the SPSS information above to construct a 95% confidence interval for the mean participation rate for all companies that have 401(k) plans.
b. Interpret the interval in the context of this problem.
c. What assumption is necessary to ensure the validity of this confidence interval?
d. If the company that conducted the sample has a 71% participation rate, can it safely conclude that its rate is below the population mean rate for all companies with 401(k) plans? Explain.
e. If in the data set the 60% had been 80%, how would the center and width of the confidence interval you constructed in part a be affected?
4. Suppose you have selected a random sample of n = 5 measurements from a normal distribution. Compare the standard normal z values with the corresponding t values if you were forming the following confidence intervals:
a. 80% confidence interval
b. 90% confidence interval
c. 95% confidence interval
d. 98% confidence interval
e. 99% confidence interval
5. The following random sample was selected from a normal distribution: 4, 6, 3, 5, 9, 3.
a. Construct a 90% confidence interval for the population mean µ
b. Construct a 95% confidence interval for the population mean µ.
c. Construct a 99% confidence interval for the population mean µ
d. Assume that the sample mean and sample standard deviations remain exactly the same as those you just calculated but that they are based on a sample of n = 25 observations rather than n = 6 observations. Repeat parts a – c. What I the effect of increasing the sample size on the width of the confidence intervals?
Increasing the sample size decreases the width of the confidence interval.
6. The following is a 90% confidence interval for p: (0.26, 0.54). How large was the sample used to construct this interval?
7. 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. Suppose a concrete-block manufacturer samples nine blocks per hour and tests the breaking strength of each. During one-hour’s test, the mean and standard deviation are 985.6 pounds per square inch (psi) and 22.9 psi, respectively.
a. Construct a 99% confidence interval for the mean breaking strength of blocks produced during the hour in which the sample was selected.
b. The process is to be considered “out of control” if the mean strength differs from 1,000 psi. What would you conclude based on the confidence interval constructed in part a?
c. Repeat parts a and b using a 90% confidence interval.
d. 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. Which interval, the 99% or 90% confidence interval, is more appropriate for making the decision? Explain.
e. Which assumptions are necessary to ensure the validity of the confidence intervals?
8. Answer each of the following:
a. Which hypothesis, the null or the alternative, is the status-quo hypothesis?
Which is the research hypothesis?
b. Which element of a test of hypothesis is used to decide whether to reject the null hypothesis in favor of the alternative hypothesis?
c. What is the level of significance called of a test of hypothesis?
d. What is the difference between Type I and Type II errors in hypothesis testing? How do a and b relate to Type I and Type II errors?
e. List the four possible results of the combinations of decisions and true states of nature for a test of hypothesis?
9. A random sample of 100 observations from a population with standard deviation 60 yielded a sample mean of 110.
a. Test the null hypothesis that µ = 100 against the alternative hypothesis that µ > 100 using α = 0.05. Interpret the results of the test.
b. Test the null hypothesis that µ = 100 against the alternative hypothesis that µ ≠ 100 using α = 0.05. Interpret the results of the test.
c. Compare the results of the two tests you conducted. Explain why the results differ.
10. A company has devised a new toner cartridge for its laser jet home/office printer that it believes has a longer lifetime (on average) than the one currently being produced. To investigate its length of life, 225 of the new cartridges were tested by counting the number of high-quality printed pages each was able to produce. The sample mean and standard deviation were determined to be 1,511.4 pages and 35.7 pages, respectively. The historical average lifetime for cartridges produced by the current process is 1,502.5 pages; the historical standard deviation is 97.3 pages.
a. What are the appropriate null and alternative hypotheses to test whether the mean lifetime of the new cartridges exceeds that of the old cartridges?
b. Use α = 0.05 to conduct the test in part a. Do the new cartridges have an average lifetime that is statistically significantly longer than the cartridges currently in production?
c. Does the difference in average lifetimes appear to be of practical significance from the perspective of the consumer? Explain.
d. Should the apparent decrease in the standard deviation in lifetimes associated with the new cartridges be viewed as an improvement over the old cartridges?
11. For each α and observed significance level (p-value) pair, indicate whether the null hypothesis would be rejected.
a. α= .05, p-value = .10
b. α= .10, p-value = .05
c. α= .01, p-value = .001
d. α= .025, p-value = .05
e. α= .10, p-value = .45
12. A random sample of 50 consumers taste tested a new snack food. Their responses were coded (0: do not like; 1: like, 2: indifferent) and recorded below:
a. Test H0: p = 0.5 against Ha: p > 0.5, where p is the proportion of customers who do not like the snack food. Use α = 0.10.
b. Find the observed significance level of your test.
1 0 0 1 2 0 1 1 0 0
0 1 0 2 0 2 2 0 0 1
1 0 0 0 0 1 0 2 0 0
0 1 0 0 1 0 0 1 0 1
0 2 0 0 1 1 0 0 0 1
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