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Chapter 9: Problem 85

Make the following hypothesis tests about \(p\). a. \(H_{0}: p=.45, \quad H_{1}: p \neq .45, \quad n=100, \quad \hat{p}=.49, \quad \alpha=.10\) b. \(H_{0}=p=.72, \quad H_{1}: p<.72, \quad n=700, \quad \hat{p}=.64, \quad \alpha=.05\) c. \(H_{0}=p=.30, \quad H_{1}: p>.30, \quad n=200, \quad \hat{p}=.33, \quad \alpha=.01\)

Short Answer

Expert verified
The solution to this exercise is determined by the final test result of comparing p-value to \(\alpha\). Therefore, after running the steps for each part, we will either reject or fail to reject the null hypothesis, \(H_0\).

Step by step solution

01

Establish Hypotheses and Significance Level

First, define the null hypothesis \(H_{0}\) and the alternative hypothesis \(H_{1}\). Also note the level of significance, \(\alpha\). For the given exercise in part a, we have \(H_{0}: p=.45, H_{1}: p \neq .45\) at \(\alpha=.10\). Since \(H_{1}\) does not equal to .45, this is a two-tailed test.
02

Calculate Standard Deviation and Test Statistic

Next, calculate the standard deviation using \(\sqrt{(p(1-p))/n}\) and the test statistic, \(Z\), using \((\hat{p}-p)/\text{standard deviation}\). Let us calculate these values for part a using \(p = .45, \hat{p} = .49\) and \(n = 100\).
03

Find p-value

Find the p-value associated with the test statistic from the Z-table. It's important to remember that for two-tailed tests, you'll need to multiply the p-value by 2.
04

Compare p-value to \(\alpha\)

Compare the calculated p-value with the given level of significance, \(\alpha\). If the p-value is less than \(\alpha\), we reject the null hypothesis, and vice versa.
05

Step 5, 6, 7 and 8: Repeat Steps for parts b and c

Repeat the same process for parts b and c while considering respectively the directionality of the tests. For part b, we have \(H_{0}=p=.72\), \(H_{1}: p<.72\), indicating a left-tailed test, while part c is a right-tailed test with \(H_{0}=p=.30\), \(H_{1}: p>.30\).

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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, often denoted as \(H_0\), is a foundational concept in hypothesis testing. It is the statement that there is no effect or no difference, and it serves as the default or starting point in the analysis.
  • For example, in part a of the given exercise, the null hypothesis is that the population proportion \(p\) equals 0.45, i.e., \(H_0: p = 0.45\).
  • The purpose of establishing \(H_0\) is to have a hypothesis that we can test against to determine if there is sufficient statistical evidence to suggest a change or difference.
In practice, scientists and statisticians start with the assumption that the null hypothesis is true, and they proceed to test it through data analysis. Only if the data strongly contradict \(H_0\), rejecting it may be considered.
Alternative Hypothesis
The alternative hypothesis, symbolized as \(H_1\) or \(H_a\), is the statement that stands in opposition to the null hypothesis. It represents the claim or outcome that we aim to support with evidence collected during testing.
  • In part b of the exercise, the alternative hypothesis is \(H_1: p < 0.72\), which indicates that we are checking if the population proportion \(p\) is indeed less than 0.72.
  • Unlike the null hypothesis, the alternative hypothesis can exhibit several forms: it might suggest a difference (two-tailed), an increase (right-tailed), or a decrease (left-tailed).
The alternative hypothesis is critical as it guides the choice of testing method and the decision criteria, specifically in determining the directionality of the test.
P-Value
The p-value plays a crucial role in statistical hypothesis testing as it helps to determine the strength of the evidence against the null hypothesis. It indicates the probability of observing results at least as extreme as those obtained, given that the null hypothesis is true.
  • In part c of the exercise, the calculated p-value helps to decide if \(H_0: p = 0.30\) can be rejected in favor of \(H_1: p > 0.30\).
  • A smaller p-value (typically less than 0.05 for common tests) suggests stronger evidence against the null hypothesis, prompting its rejection.
Understanding the p-value is essential for making data-driven decisions and assessing the likelihood of observed data under assumed conditions.
Level of Significance
The level of significance, represented by \(\alpha\), is a predetermined threshold for making decisions in hypothesis testing. It defines the maximum probability of committing a Type I error, which is the error of rejecting a true null hypothesis.
  • In part a of the exercise, \(\alpha = 0.10\) means there is a 10% risk of incorrectly rejecting the null hypothesis if it's true.
  • Commonly used significance levels include 0.05, 0.01, and 0.10.
Choosing the level of significance depends on the context of the test and the potential consequences of errors. At a lower \(\alpha\), the test becomes more conservative, reducing the chance of falsely rejecting a true null hypothesis but potentially increasing the chance of failing to reject a false one.

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Most popular questions from this chapter

Consider the null hypothesis \(H_{0}: p=.65 .\) Suppose a random sample of 1000 observations is taken to perform this test about the population proportion. Using \(\alpha=.05\), show the rejection and nonrejection regions and find the critical value(s) of \(z\) for a a. left-tailed test b. two-tailed test c. right-tailed test

According to a New York Times/CBS News poll conducted during June \(24-28,2011,55 \%\) of the American adults polled said that owning a home is a very important part of the American Dream (The New York Times, June 30,2011 ). Suppose this result was true for the population of all American adults in \(2011 .\) In a recent poll of 1800 American adults, \(61 \%\) said that owning a home is a very important part of the American Dream. Perform a hypothesis test to determine whether it is reasonable to conclude that the percentage of all American adults who currently hold this opinion is higher than \(55 \%\). Use a \(2 \%\) significance level, and use both the \(p\) -value and the critical-value approaches.

Consider \(H_{0}=\mu=40\) versus \(H_{1}: \mu>40 .\) a. A random sample of 64 observations taken from this population produced a sample mean of 43 and a standard deviation of \(5 .\) Using \(\alpha=.025\), would you reject the null hypothesis? b. Another random sample of 64 observations taken from the same population produced a sample mean of 41 and a standard deviation of \(7 .\) Using \(\alpha=.025\), would you reject the null hypothesis? Comment on the results of parts a and \(\mathrm{b}\).

Two years ago, \(75 \%\) of the customers of a bank said that they were satisfied with the services provided by the bank. The manager of the bank wants to know if this percentage of satisfied customers has changed since then. She assigns this responsibility to you. Briefly explain how you would conduct such a tes

Explain which of the following is a two-tailed test, a left-tailed test, or a right-tailed test. a. \(H_{0}: \mu=12, \quad H_{1}: \mu<12 \quad\) b. \(H_{0}: \mu \leq 85, \quad H_{1}: \mu>85\) c. \(H_{0}: \mu=33, \quad H_{1}: \mu \neq 33\) Show the rejection and nonrejection regions for each of these cases by drawing a sampling distribution curve for the sample mean, assuming that it is normally distributed.
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