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Chapter 10: Problem 42

Suppose we are testing the hypothesis \(H_{0}: p=0.65\) versus \(H_{1}: p \neq 0.65\) and we find the \(P\) -value to be \(0.02 .\) Explain what this means. Would you reject the null hypothesis? Why?

Short Answer

Expert verified
Reject the null hypothesis because the P-value (0.02) is less than 0.05.

Step by step solution

01

- Understand the Hypotheses

Identify the null and alternative hypotheses. The null hypothesis (\(H_{0}\)) states that the population proportion \(p\) is 0.65. The alternative hypothesis (\(H_{1}\)) states that the population proportion \(p\) is not equal to 0.65.
02

- Interpret the P-value

The \(P\)-value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is correct. Here, the \(P\)-value is 0.02.
03

- Decision Rule

Determine the significance level (\(伪\)). Commonly used significance levels are 0.05 or 0.01. Compare the \(P\)-value with the significance level. If the \(P\)-value is less than the significance level, reject the null hypothesis.
04

- Make the Decision

Given that the \(P\)-value (0.02) is less than the common significance level of 0.05, we reject the null hypothesis. This means there is sufficient evidence to suggest that the population proportion is different from 0.65.

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Key Concepts

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

null hypothesis
The null hypothesis, denoted as \(H_{0}\), is a statement that there is no effect or no difference, and it serves as the starting point for our hypothesis test. In this problem, the null hypothesis is \(H_{0}: p = 0.65\). This means we initially assume that the population proportion is 0.65. The null hypothesis is what we seek to test against the alternative hypothesis, and it is presumed true until evidence suggests otherwise.
Phrasing it another way, the null hypothesis is like a default position; it is what we accept as true until our test provides sufficient evidence to believe otherwise.
alternative hypothesis
The alternative hypothesis, represented as \(H_{1}\), proposes an alternative to the null hypothesis. It is what we want to prove. In this context, the alternative hypothesis is \(H_{1}: p eq 0.65\), which means the population proportion is not 0.65. The alternative hypothesis is what we consider if the evidence contradicts the null hypothesis.
Unlike the null hypothesis, which is assumed correct until proven wrong, the alternative hypothesis is only accepted if the data provides strong evidence against \(H_{0}\).
P-value
The P-value measures the strength of the evidence against the null hypothesis. It answers the question: How likely is it that we would observe such an extreme test result, assuming the null hypothesis is true? In this exercise, the P-value is 0.02.
Lower P-values indicate stronger evidence against the null hypothesis. Since our P-value here is 0.02, it means there is a 2% chance of obtaining a result as extreme as the observed one, assuming \(H_{0}\) is correct.
significance level
The significance level, denoted by \(\alpha\), is the threshold at which we decide whether the P-value is low enough to reject the null hypothesis. Common choices for significance levels include 0.05 (5%) and 0.01 (1%).
The significance level is chosen before conducting the test and represents the probability of rejecting the null hypothesis when it is actually true (Type I error). In our problem, we compare the P-value of 0.02 to a significance level of 0.05. Since 0.02 < 0.05, we reject the null hypothesis.
statistical decision
A statistical decision involves comparing the P-value to the significance level and deciding whether to reject or fail to reject the null hypothesis. Here, our P-value is 0.02.
Given a common significance level of 0.05, we find that 0.02 < 0.05. Thus, we reject the null hypothesis, meaning there is sufficient evidence to support the alternative hypothesis. In this context, we have enough evidence to suggest the population proportion is different from 0.65.

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Most popular questions from this chapter

Explain the term power of the test.

In December \(2001,38 \%\) of adults with children under the age of 18 reported that their family ate dinner together seven nights a week. In a recent poll, 403 of 1122 adults with children under the age of 18 reported that their family ate dinner together seven nights a week. Has the proportion of families with children under the age of 18 who eat dinner together seven nights a week decreased? Use the \(\alpha=0.05\) significance level.

A machine fills bottles with 64 fluid ounces of liquid. The quality-control manager determines that the fill levels are normally distributed with a mean of 64 ounces and a standard deviation of 0.42 ounce. He has an engineer recalibrate the machine in an attempt to lower the standard deviation. After the recalibration, the quality-control manager randomly selects 19 bottles from the line and determines that the standard deviation is 0.38 ounce. Is there less variability in the filling machine? Use 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.

In an American Animal Hospital Association survey, \(37 \%\) of respondents stated that they talk to their pets on the telephone. A veterinarian found this result hard to believe, so he randomly selected 150 pet owners and discovered that 54 of them spoke to their pet on the telephone. Does the veterinarian have the right to be skeptical? Use a 0.05 level of significance.
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