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Chapter 9: Problem 13

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, by rejecting the null hypothesis, we are not stating that the alternative hypothesis is true. We are merely suggesting that there's enough evidence to favor the alternative hypothesis over the null.

Step by step solution

01

Understanding Null and Alternative Hypothesis

In hypothesis testing, there are two competing hypotheses: the null hypothesis, often denoted as \(H_0\), and the alternative hypothesis, denoted as \(H_1\) or \(H_a\). The null hypothesis represents a statement of no effect or no difference. The alternative hypothesis represents, typically, the research hypothesis, or the statement of a new effect or difference.
02

Understanding Rejection of Null Hypothesis

When we reject the null hypothesis, we're saying that our sample provides enough evidence to suggest the null hypothesis is unlikely and that something else is going on. But this does not necessarily mean the alternative hypothesis is correct.
03

Final Interpretation

By rejecting the null hypothesis, we're suggesting that there's enough statistical evidence to favor the alternative hypothesis over the null. However, we're not stating the alternative hypothesis is true. This is because hypothesis testing is based on probability and statistics, and thus we cannot make absolute claims.

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Most popular questions from this chapter

Consider the following null and alternative hypotheses: $$ H_{0}: p=.82 \text { versus } H_{1}: p \neq .82 $$ A random sample of 600 observations taken from this population produced a sample proportion of \(.86 .\) a. If this test is made at the \(2 \%\) significance level, would you reject the null hypothesis? Use the critical-value approach. b. What is the probability of making a Type I error in part a? c. Calculate the \(p\) -value for the test. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=.025 ?\) What if \(\alpha=.005 ?\)

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.

More and more people are abandoning national brand products and buying store brand products to save money. The president of a company that produces national brand coffee claims that \(40 \%\) of the people prefer to buy national brand coffee. A random sample of 700 people who buy coffee showed that 259 of them buy national brand coffee. Using \(\alpha=.01\), can you conclude that the percentage of people who buy national brand coffee is different from \(40 \%\) ? Use both approaches to make the test.

Professor Hansen believes that some people have the ability to predict in advance the outcome of a spin of a roulette wheel. He takes 100 student volunteers to a casino. The roulette wheel has 38 numbers, each of which is equally likely to occur. Of these 38 numbers, 18 are red, 18 are black, and 2 are green. Each student is to place a serics of five bets, choosing either a red or a black number before each spin of the wheel. Thus, a student who bets on red has an \(18 / 38\) chance of winning that bet. The same is true of betting on black. a. Assuming random guessing, what is the probability that a particular student will win all five of his or her bets? b. Suppose for each student we formulate the hypothesis test \(H_{0}:\) The student is guessing \(H_{1}:\) The student has some predictive ability Suppose we reject \(H_{0}\) only if the student wins all five bets. What is the significance level? c. Suppose that 2 of the 100 students win all five of their bets. Professor Hansen says, "For these two students we can reject \(H_{0}\) and conclude that we have found two students with some ability to predict." What do you make of Professor Hansen's conclusion?

A May 8,2008 , report on National Public Radio (www.npr.org) noted that the average age of firsttime mothers in the United States is slightly higher than 25 years. Suppose that a recently taken random sample of 57 first-time mothers from Missouri produced an average age of \(23.90\) years and that the population standard deviation is known to be \(4.80\) years. a. Find the \(p\) -value for the test of hypothesis with the alternative hypothesis that the current mean age of all first-time mothers in Missouri is less than 25 years. Will you reject the null hypothesis at \(\alpha=.025\) ? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.025\).
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