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Chapter 10: Problem 41

Suppose we are testing the hypothesis \(H_{0}: p=0.3\) versus \(H_{1}: p>0.3\) and we find the \(P\) -value to be \(0.23 .\) Explain what this means. Would you reject the null hypothesis? Why?

Short Answer

Expert verified
Do not reject the null hypothesis because the P-value of 0.23 is greater than the significance level of 0.05.

Step by step solution

01

Understand the Hypotheses

The null hypothesis (H_0) states that the population proportion ( p) is 0.3 and the alternative hypothesis ( H_1) states that the population proportion ( p) is greater than 0.3.
02

Identify the P-value

The P-value for the test is given as 0.23. The P-value measures the probability of obtaining a test statistic at least as extreme as the one obtained, assuming the null hypothesis is true.
03

Compare the P-value to the Significance Level

Typically, we compare the P-value to a significance level ( alpha ), often set at 0.05. If P-value is less than or equal to alpha , we reject the null hypothesis.
04

Make a Decision

Since 0.23 is greater than 0.05, we do NOT reject the null hypothesis. The P-value is not small enough to provide strong evidence against H_0 .

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

P-value interpretation
The P-value plays a crucial role in hypothesis testing. It indicates the probability of obtaining a test statistic at least as extreme as the one observed, given that the null hypothesis is true. For example, a P-value of 0.23 means there is a 23% chance of observing the given or a more extreme result under the assumption that the null hypothesis holds. A higher P-value, like 0.23, suggests that the observed result is relatively likely when the null hypothesis is true.
null hypothesis
The null hypothesis, usually denoted as H_0, is a statement that there is no effect or no difference. It serves as the starting point for testing. In our example, the null hypothesis is that the population proportion (p) is 0.3. This hypothesis remains 'innocent until proven guilty' and is only rejected if the data provides strong enough evidence against it. We assume H_0 is true when calculating probabilities like the P-value.
significance level
The significance level, denoted by 伪 (alpha), is the threshold that determines when we reject the null hypothesis. Commonly, 伪 is set at 0.05, which corresponds to a 5% risk of rejecting the null hypothesis when it is actually true (Type I error). If the P-value is less than or equal to 伪, we reject the null hypothesis. For example, if 伪 = 0.05 and the observed P-value is 0.23, we do not reject H_0 because 0.23 > 0.05.
alternative hypothesis
The alternative hypothesis, denoted as H_1, is what you want to support. It usually represents a new effect or difference. In our example, H_1 states that the population proportion (p) is greater than 0.3. While the null hypothesis assumes no change or effect, the alternative hypothesis suggests a specific direction or magnitude of change. If evidence strong enough to reject H_0 is found, we provide support for H_1. However, a high P-value (like 0.23) does not offer such strong evidence, so we do not reject H_0.

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Most popular questions from this chapter

In Problems \(9-14,\) the null and alternative hypotheses are given. Determine whether the hypothesis test is lefi-tailed, right-tailed, or two-tailed. What parameter is being tested? \(H_{0}: p=0.2\) \(H_{1}: p<0.2\)

To test \(H_{0}: \mu=20\) versus \(H_{1}: \mu<20,\) a simple random sample of size \(n=18\) is obtained from a population that is known to be normally distributed. (a) If \(\bar{x}=18.3\) and \(s=4.3,\) compute the test statistic. (b) Draw a \(t\) -distribution with the area that represents the \(P\) -value shaded. (c) Approximate and interpret the \(P\) -value. (d) If the researcher decides to test this hypothesis at the \(\alpha=0.05\) level of significance, will the researcher reject the null hypothesis? Why?

A manufacturer of high-strength, lowalloy steel beams requires that the standard deviation of yield strength not exceed 7000 pounds per square inch (psi). The quality-control manager selected a sample of 20 steel beams and measured their yield strength. The standard deviation of the sample was 7500 psi. Assume that yield strengths are normally distributed. Does the evidence suggest that the standard deviation of yield strength exceeds 7000 psi at the \(\alpha=0.01\) level of significance?

Simulation Simulate drawing 100 simple random samples of size \(n=15\) from a population that is normally distributed with mean 100 and standard deviation 15 . (a) Test the null hypothesis \(H_{0}: \mu=100\) versus \(H_{1}: \mu \neq 100\) for each of the 100 simple random samples. (b) If we test this hypothesis at the \(\alpha=0.05\) level of significance, how many of the 100 samples would you expect to result in a Type I error? (c) Count the number of samples that lead to a rejection of the null hypothesis. Is it close to the expected value determined in part (b)? (d) Describe how we know that a rejection of the null hypothesis results in making a Type I error in this situation.

To test \(H_{0}: \mu=100\) versus \(H_{1}: \mu \neq 100,\) a simple random sample of size \(n=23\) is obtained from a population that is known to be normally distributed. (a) If \(\bar{x}=104.8\) and \(s=9.2,\) compute the test statistic. (b) If the researcher decides to test this hypothesis at the \(\alpha=0.01\) level of significance, determine the critical values. (c) Draw a \(t\) -distribution that depicts the critical region. (d) Will the researcher reject the null hypothesis? Why? (e) Construct a \(99 \%\) confidence interval to test the hypothesis.
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