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Chapter 9: Problem 74

Explain when a sample is large enough to use the normal distribution to make a test of hypothesis about the population proportion.

Short Answer

Expert verified
A sample is large enough to use the normal distribution for hypothesis testing about the population proportion when conditions \( n \cdot p \geq 10 \) and \( n \cdot (1 - p) \geq 10 \) are met, where \( n \) is the sample size and \( p \) is the population proportion.

Step by step solution

01

Understanding Central Limit Theorem

The Central Limit Theorem (CLT) is a statistical theory states that, given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population, and the distribution of the sample means will approach a normal distribution pattern. This is the physical foundation which allows us to use normal distribution.
02

Conditions for Using Normal Distribution

To use the normal distribution for hypothesis testing about population proportion, the sample size must be large enough. The two conditions based on Binomial Distribution that must meet: 1. \( n \cdot p \geq 10 \)2. \( n \cdot (1 - p) \geq 10 \)where \( n \) is the sample size and \( p \) is the population proportion. If these conditions are met, then the distribution of the sample proportion \( \hat{p} \) is approximately normally distributed.
03

Relating to Hypothesis Testing

In terms of hypothesis testing, the above conditions enable us to use normal distribution to perform Z-tests to make inference about the population proportion. That includes drawing conclusions if the observed sample proportion is statistically significantly different from a hypothesized population proportion.

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Most popular questions from this chapter

Lazurus Steel Corporation produces iron rods that are supposed to be 36 inches long. The machine that makes these rods does not produce each rod exactly 36 inches long. The lengths of the rods are normally distributed, and they vary slightly. It is known that when the machine is working properly, the mean length of the rods is 36 inches. The standard deviation of the lengths of all rods produced on this machine is always equal to \(.035\) inch. The quality control department at the company takes a sample of 20 such rods every week, calculates the mean length of these rods, and tests the null hypothesis, \(\mu=36\) inches, against the alternative hypothesis, \(\mu \neq 36\) inches. If the null hypothesis is rejected, the machine is stopped and adjusted. A recent sample of 20 rods produced a mean length of \(36.015\) inches. a. Calculate the \(p\) -value for this test of hypothesis. Based on this \(p\) -value, will the quality control inspector decide to stop the machine and adjust it if he chooses the maximum probability of a Type I error to be \(.02 ?\) What if the maximum probability of a Type I error is .10? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.02 .\) Does the machine need to be adjusted? What if \(\alpha=.10\) ?

According to an article in the Chicago Sun-Times (http://jachakim.com/articles/lifestyles/engage.htm), the average length of an engagement that results in a marriage in the United States is 14 months. Suppose that a random sample of 99 recently married Canadian couples had an average engagement length of \(12.84\) months, with a sample standard deviation of \(4.52\) months. Does the sample information support the alternative hypothesis that the average engagement length in Canada is different from 14 months, the average length in the United States? Use a \(10 \%\) significance level. Use both the \(p\) -value approach and the critical-value approach.

Alpha Airlines claims that only \(15 \%\) of its flights arrive more than 10 minutes late. Let \(p\) be the proportion of all of Alpha's flights that arrive more than 10 minutes late. Consider the hypothesis test $$ H_{0}: p \leq .15 \text { versus } H_{1}: p>.15 $$ Suppose we take a random sample of 50 flights by Alpha Airlines and agree to reject \(H_{0}\) if 9 or more of them arrive late. Find the significance level for this test.

A statistician performs the test \(H_{0}: \mu=15\) versus \(H_{1}: \mu \neq 15\) and finds the \(p\) -value to be \(.4546\). a. The statistician performing the test does not tell you the value of the sample mean and the value of the test statistic. Despite this, you have enough information to determine the pair of \(p\) -values associated with the following alternative hypotheses. i. \(H_{1}: \mu<15\) ii. \(H_{1}: \mu>15\) Note that you will need more information to determine which \(p\) -value goes with which alternative. Determine the pair of \(p\) -values. Here the value of the sample mean is the same in both cases. b. Suppose the statistician tells you that the value of the test statistic is negative. Match the \(p\) -values with the alternative hypotheses. Note that the result for one of the two alternatives implies that the sample mean is not on the same side of \(\mu=15\) as the rejection region. Although we have not discussed this scenario in the book, it is important to recognize that there are many real-world scenarios in which this type of situation does occur. For example, suppose the EPA is to test whether or not a company is exceeding a specific pollution level. If the average discharge level obtained from the sample falls below the threshold (mentioned in the null hypothesis), then there would be no need to perform the hypothesis test.

Thirty percent of all people who are inoculated with the current vaccine used to prevent a disease contract the disease within a year. The developer of a new vaccine that is intended to prevent this disease wishes to test for significant evidence that the new vaccine is more effective. a. Determine the appropriate null and alternative hypotheses. b. The developer decides to study 100 randomly selected people by inoculating them with the new vaccine. If 84 or more of them do not contract the disease within a year, the developer will conclude that the new vaccine is superior to the old one. What significance level is the developer using for the test? c. Suppose 20 people inoculated with the new vaccine are studied and the new vaccine is concluded to be better than the old one if fewer than 3 people contract the disease within a year. What is the significance level of the test?
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