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Chapter 9: Problem 11

By rejecting the null hypothesis in a test of hypothesis example, are you stating that the alternative hypothesis is true?

Short Answer

Expert verified
No, rejecting the null hypothesis in a test does not necessarily state that the alternative hypothesis is true. It simply indicates that there is enough statistical evidence to consider the null hypothesis unlikely, making the alternative hypothesis more plausible. However, it doesn't confirm the alternative hypothesis outright.

Step by step solution

01

Understanding Hypothesis Testing

Hypothesis testing is a statistical method that tests the validity of a claim or proposition about a population parameter. A null hypothesis is an assumption or a statement about a parameter that we try to reject using our sample data. The alternative hypothesis is the complement of the null hypothesis, an assertion of what will be accepted if the sample evidence fails to support the null hypothesis.
02

Why do we reject a null hypothesis?

We reject a null hypothesis because there is enough statistical evidence to show that it is unlikely to be true. This could be said when our test statistic is in the critical region defined by the rejection area of the p-value (usually less than a significance level of 0.05).
03

Does rejecting the null hypothesis prove the alternative hypothesis?

Rejecting the null hypothesis does not necessarily prove that the alternative hypothesis is true. It simply means there's enough evidence to suggest that the null hypothesis is highly unlikely, and thus, the alternative hypothesis is more plausible. But this doesn't confirm it outright. It's always possible that future data may support the null hypothesis. Thus, we use the term 'fail to reject' instead of 'accept' for the null hypothesis. When we reject the null hypothesis, it's safer to say that the test supports the alternative hypothesis, rather than it proves it.

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Most popular questions from this chapter

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.

The manager of a service station claims that the mean amount spent on gas by its customers is \(\$ 15.90\) per visit. You want to test if the mean amount spent on gas at this station is different from \(\$ 15.90\) per visit. Briefly explain how you would conduct this test when \(\sigma\) is not known.

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 that is 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 altemative 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?

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. \(H_{0}: \mu=38, \quad H_{1}: \mu>38, \quad n=35, \quad \bar{x}=40.25, \quad \sigma=7.2\)
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