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Chapter 9: Problem 49

For each of the following examples of tests of hypothesis about \(\mu\), show the rejection and nonrejection regions on the \(t\) distribution curve. a. A two-tailed test with \(\alpha=.01\) and \(n=15\) b. A left-tailed test with \(\alpha=.005\) and \(n=25\) c. A right-tailed test with \(\alpha=.025\) and \(n=22\)

Short Answer

Expert verified
Rejection regions for the given tests are: a) \(t< -2.977\) and \(t > 2.977\), b) \(t < -2.492\), and c) \(t > 2.080\). Beyond these t-values, the null hypothesis would be rejected.

Step by step solution

01

Hypothesis Testing Overview

Hypothesis testing is a statistical method that is used in making statistical decisions using experimental data. It involves the statement of a null hypothesis, and the selection of a level of significance. The hypothesis is assumed to be true until statistical evidence in the form of a hypothesis test indicates otherwise.
02

Two-Tailed Test

A two-tailed test with an alpha level of .01 and sample size of 15 means we're checking for an extreme value in either tail of the distribution. We reject the null hypothesis if our test statistic is in the top 0.005 (i.e., \(\alpha/2\)) or the bottom 0.005. To determine the critical value, we also need to calculate the degrees of freedom, which is \(n-1\) or 14. Looking up these values in a t-table, we have a critical value of about \(\pm2.977\). Any t-score beyond these values falls in the rejection region.
03

Left-Tailed Test

A left-tailed test with an alpha level of .005 and n = 25 implies we're checking for an extreme value only in the left tail of the distribution. We reject the null hypothesis if our test statistic is in the bottom 0.005. Degrees of freedom are 24. From a t-table, the critical value is about -2.492, so any t-score below this is in the rejection region.
04

Right-Tailed Test

Right-tailed test with an alpha level of .025 and n = 22 implies we're checking for an extreme value only in the right tail of the distribution. We reject the null hypothesis if our test statistic is in the top 0.025. Degrees of Freedom this time are 21. From the t-table, the critical value is roughly 2.080, hence any t-score greater than this falls in the rejection region.

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Most popular questions from this chapter

Explain which of the following is a two-tailed test, a left-tailed test, or a right-tailed test. a. \(H_{0}: \mu=45, \quad H_{1}: \mu>45\) b. \(H_{0}: \mu=23, \quad H_{1}: \mu \neq 23\) c. \(H_{0}: \mu \geq 75, \quad H_{1}: \mu<75\)

Shulman Steel Corporation makes bearings that are supplied to other companies. One of the machines makes bearings that are supposed to have a diamcter of 4 inches. The bearings that have a diameter of either more or less than 4 inches are considered defective and are discarded. When working properly, the machine does not produce more than \(7 \%\) of bearings that are defective. The quality control inspector selects a sample of 200 bearings each week and inspects them for the size of their diameters. Using the sample proportion, the quality control inspector tests the null hypothesis \(p \leq .07\) against the alternative hypothesis \(p \geq\) 07, where \(p\) is the proportion of bearings that are defective. He always uses a \(2 \%\) significance level. If the null hypothesis is rejected, the machine is stopped to make any necessary adjustments. One sample of 200 bearings taken recently contained 22 defective bearings. a. Using the \(2 \%\) significance level, will you conclude that the machine should be stopped to make necessary adjustments? b. Perform the test of part a using a \(1 \%\) significance level. Is your decision different from the one in part a?

A random sample of 80 observations produced a sample mean of \(86.50\). Find the critical and observed values of \(z\) for each of the following tests of hypothesis using \(\alpha=.10 .\) The population standard deviation is known to be \(7.20\). a. \(H_{0}: \mu=91\) versus \(H_{1}: \mu \neq 91\) b. \(H_{0}: \mu=91 \quad\) versus \(\quad H_{1}: \mu<91\)

The past records of a supermarket show that its customers spend an average of \(\$ 95\) per visit at this store. Recently the management of the store initiated a promotional campaign according to which each customer receives points based on the total money spent at the store, and these points can be used to buy products at the store. The management expects that as a result of this campaign, the customers should be encouraged to spend more money at the store. To check whether this is true, the manager of the store took a sample of 14 customers who visited the store. The following data give the money (in dollars) spent by these customers at this supermarket during their visits. \(\begin{array}{rrrrrrr}109.15 & 136.01 & 107.02 & 116.15 & 101.53 & 109.29 & 110.79 \\ 94.83 & 100.91 & 97.94 & 104.30 & 83.54 & 67.59 & 120.44\end{array}\) Assume that the money spent by all customers at this supermarket has a normal distribution. Using the \(5 \%\) significance level, can you conclude that the mean amount of money spent by all customers at this supermarket after the campaign was started is more than \(\$ 95 ?\) (Hint: First calculate the sample mean and the sample standard deviation for these data using the formulas learned in Sections \(3.1 .1\) and \(3.2 .2\) of Chapter \(3 .\) Then make the test of hypothesis about \(\mu .\) )

In a 2009 nonscientific poll on the Web site of the Daily Gazette of Schenectady, New York, readers were asked the following question: "Are you less inclined to buy a General Motors or Chrysler vehicle now that they have filed for bankruptcy?" Of the respondents, \(56.1 \%\) answered "Yes" (http://www. dailygazette.com/polls/2009/jun/Bankruptcy/). In a recent survey of 1200 adult Americans who were asked the same question, 615 answered "Yes." Can you reject the null hypothesis at the \(1 \%\) significance level in favor of the alternative that the percentage of all adult Americans who are less inclined to buy a General Motors or Chrysler vehicle since the companies filed for bankruptcy is different from \(56.1 \%\) ? Use both the \(p\) -value and the critical-value approaches.
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