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Chapter 7: Q25 E (page 403)
In a test of \({H_0}:\mu = 100\) against \({H_a}:\mu \ne 100\), the sample data yielded the test statistic z = 2.17. Find the p-value for the test.
The p-value for the hypothesis test is 0.030.
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We reject the null hypothesis when the test statistic falls in the rejection region, but we do not accept the null hypothesis when the test statistic does not fall in the rejection region. Why?
Latex allergy in health care workers. Refer to the Current Allergy & Clinical Immunology (March 2004) study of n = 46 hospital employees who were diagnosed with a latex allergy from exposure to the powder on latex gloves, Exercise 6.112 (p. 375). The number of latex gloves used per week by the sampled workers is summarized as follows: \(\bar x = 19.3\) and s = 11.9. Let \(\mu \) represent the mean number of latex gloves used per week by all hospital employees. Consider testing \({H_0}:\mu = 20\) against \({H_a}:\mu < 20.\)
a. Give the rejection region for the test at a significance level of \(\alpha = 0.01.\)
Suppose a random sample of 100 observations from a binomial population gives a value of \(\hat p = .63\) and you wish to test the null hypothesis that the population parameter p is equal to .70 against the alternative hypothesis that p is less than .70.
a. Noting that\(\hat p = .63\) what does your intuition tell you? Does the value of \(\hat p\) appear to contradict the null hypothesis?
Manufacturers that practice sole sourcing. If a manufacturer (the vendee) buys all items of a particular type from a particular vendor, the manufacturer is practicing sole sourcing (Schonberger and Knod, Operations Management, 2001). As part of a sole-sourcing arrangement, a vendor agrees to periodically supply its vendee with sample data from its production process. The vendee uses the data to investigate whether the mean length of rods produced by the vendor's production process is truly 5.0 millimetres (mm) or more, as claimed by the vendor and desired by the vendee.
a. If the production process has a standard deviation of .01 mm, the vendor supplies n = 100 items to the vendee, and the vendee uses a = .05 in testing H0: m = 5.0 mm against Ha: m < 5.0 mm, what is the probability that the vendee's test will fail to reject the null hypothesis when in fact m = 4.9975 mm? What is the name given to this Type of error?
b. Refer to part a. What is the probability that the vendee's test will reject the null hypothesis when m = 5.0? What is the name given to this Type of error?
c. What is the power of the test to detect a departure of .0025 mm below the specified mean rod length of 5.0 mm?
Suppose you are interested in conducting the statistical test of \({H_0}:\mu = 255\) against \({H_a}:\mu > 225\), and you have decided to use the following decision rule: Reject H0 if the sample mean of a random sample of 81 items is more than 270. Assume that the standard deviation of the population is 63.
a. Express the decision rule in terms of z.
b. Find \(\alpha \), the probability of making a Type I error by using this decision rule.
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