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Chapter 9: Problem 56

Consider the null hypothesis \(H_{0}: p=.65 .\) Suppose a random sample of 1000 observations is taken to perform this test about the population proportion. Using \(\alpha=.05\), show the rejection and nonrejection regions and find the critical value(s) of \(z\) for a a. left-tailed test b. two-tailed test c. right-tailed test

Short Answer

Expert verified
The critical z-scores for a left-tailed test, a two-tailed test, and a right-tailed test are -1.645, 卤1.96, and 1.645 respectively.

Step by step solution

01

Left-tailed test

For a left-tailed test, the rejection region is the portion of the distribution below certain z-score. This z-score corresponds to the significance level \(\alpha\). To find this z-score, a solution can use the inverse cumulative distribution function (also known as quantile function). The critical z-score for a left-tailed test is thus \(z = N^{-1}(\alpha)\), where \(N^{-1}\) function is the inverse of cumulative distribution function of the normal distribution.
02

Two-tailed test

For a two-tailed test, the rejection region is the portion of the distribution that lies either below or above certain z-scores. Since the test is two-sided, we divide the significance level \(\alpha\) by 2 before finding the z-score. The z-scores corresponding to the two rejection regions are \(z = N^{-1}\left(\frac{\alpha}{2}\right)\) and \(z = -N^{-1}\left(\frac{\alpha}{2}\right)\).
03

Right-tailed test

For a right-tailed test, the rejection region is the portion of the distribution above certain z-score. This z-score corresponds to \(1-\alpha\). That is, \(z = N^{-1}(1-\alpha)\).
04

Calculating the Z-Scores

With a significance level of 0.05, the critical z-scores for these tests are: Left-tailed test: \(Z = N^{-1}(0.05) = -1.645\), Two-tailed test: \(Z = \pm N^{-1}(0.025) = \pm1.96\), Right-tailed test: \(Z = N^{-1}(0.95) = 1.645\)

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Most popular questions from this chapter

Dartmouth Distribution Warehouse makes deliveries of a large number of products to its customers. To keep its customers happy and satisfied, the company's policy is to deliver on time at least \(90 \%\) of all the orders it receives from its customers. The quality control inspector at the company quite often takes samples of orders delivered and checks to see whether this policy is maintained. A recent sample of 90 orders taken by this inspector showed that 75 of them were delivered on time. a. Using a \(2 \%\) significance level, can you conclude that the company's policy is maintained? b. What will your decision be in part a if the probability of making a Type I error is zero? Explain.

Write the null and alternative hypotheses for each of the following examples. Determine if each is a case of a two-tailed, a left-tailed, or a right-tailed test. a. To test if the mean amount of time spent per week watching sports on television by all adult men is different from \(9.5\) hours b. To test if the mean amount of money spent by all customers at a supermarket is less than \(\$ 105\) c. To test whether the mean starting salary of college graduates is higher than \(\$ 47,000\) per year d. To test if the mean waiting time at the drive-through window at a fast food restaurant during rush hour differs from 10 minutes e. To test if the mean time spent per week on house chores by all housewives is less than 30 hours

According to an article in Forbes magazine of April 3, 2014 , \(57 \%\) of students said that they did not attend the college of their first choice due to financial concerns (www.forbes.com). In a recent poll of 1600 students, 864 said that they did not attend the college of their first choice due to financial concerns. Using a \(1 \%\) significance level. perform a test of hypothesis to determine whether the current percentage of students who did not attend the college of their first choice due to financial concerns is lower than \(57 \%\). Use both the \(p\) -value and the critical-value approaches.

Find the \(p\) -value for each of the following hypothesis tests. a. \(H_{0}: \mu=23, \quad H_{1}: \mu \neq 23, \quad n=50, \quad \bar{x}=21.25, \quad \sigma=5\) b. \(H_{0}: \mu=15, \quad H_{1}: \mu<15, \quad n=80, \quad \bar{x}=13.25, \quad \sigma=5.5\) c. \(H_{0}: \mu=38, \quad H_{1}: \mu>38, \quad n=35, \quad \bar{x}=40.25, \quad \sigma=7.2\)

According to an estimate, 2 years ago the average age of all CEOs of medium- sized companies in the United States was 58 years. Jennifer wants to check if this is still true. She took a random sample of 70 such CEOs and found their mean age to be 55 years with a standard deviation of 6 years. a. Suppose that the probability of making a Type I error is selected to be zero. Can you conclude that the current mean age of all CEOs of medium-sized companies in the United States is different from 58 years? b. Using a \(1 \%\) significance level, can you conclude that the current mean age of all CEOs of medium-sized companies in the United States is different from 58 years? Use both approaches.
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