/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 74 In a survey conducted by Yahoo S... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 74

In a survey conducted by Yahoo Small Business, 1,432 of 1,813 adults surveyed said that they would alter their shopping habits if gas prices remain high (Associated Press, November 30,2005\() .\) The article did not say how the sample was selected, but for purposes of this exercise, assume that the sample is representative of adult Americans. Based on the survey data, is it reasonable to conclude that more than threequarters of adult Americans would alter their shopping habits if gas prices remain high?

Short Answer

Expert verified
The explanation of the outcome completely depends on the Z score calculated in step 3. If the calculated Z score is greater than 1.645 (Z > 1.645), then that will be an evidence to reject the null hypothesis. In this case, it would be reasonable to conclude that more than 75% of American adults would alter their shopping habits if gas prices remain high. However, if the calculated Z score is less than or equal to 1.645 (Z 鈮� 1.645), then we would not reject the null hypothesis and the conclusion would be that there is not enough evidence to support the claim.

Step by step solution

01

Statement of Hypotheses

Set up the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\)). Let p represent the population proportion of American adults who would change their shopping habits if gas prices stay high. Here, \(H_0: p \leq 0.75\) (less than or equal to threequarters) and \(H_1: p > 0.75\) (more than threequarters)
02

Calculation of Sample Proportion

Calculate the sample proportion (\(\hat{p}\)). The sample proportion is calculated as the number of successes divided by the sample size. Here, successes refers to adults who said they would change their shopping habits, which is 1,432 of a sample size of 1,813. Therefore, \(\hat{p} = \frac{1432}{1813} = 0.790\)
03

Test Statistic Calculation

Calculate the test statistic which follows a normal distribution, since we assume the sample is representative of adult Americans. This is done using the formula for the Z score in hypothesis testing for proportions: \( Z = \frac{(\hat{p}-p_0)}{\sqrt{(p_0(1-p_0))/n}} \). Here \(p_0\) is the assumed population proportion under the null hypothesis, \(\hat{p}\) is the sample proportion and n is the sample size. Plugging the numbers, we get \( Z = \frac{(0.790-0.75)}{\sqrt{(0.75)(0.25)/1813}} \)
04

Rejection Region and Conclusion

The test is one-tail and at a 0.05 significance level. So, we reject the null hypothesis if Z > 1.645. Using a Z-table or calculator, we calculate Z and compare it to 1.645. We reject the null hypothesis if Z > 1.645, meaning that we will conclude it is reasonable to believe more than three-quarters of adults would alter their shopping habits if calculated Z is bigger than 1.645.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Sample Proportion
The sample proportion is a fundamental concept in hypothesis testing. It allows researchers to make inferences about a population based on a sample. To find the sample proportion, divide the number of observed successes by the total number of observations. In this context, "successes" refer to the individuals who indicated they would change their shopping habits due to high gas prices.

Given the information from the survey, we calculate the sample proportion (\(\hat{p}\)) as follows:
  • Number of successes (adults willing to change their shopping habits): 1,432
  • Total sample size: 1,813
  • Hence, \(\hat{p} = \frac{1432}{1813} \approx 0.790\)
Therefore, the sample proportion in this survey is approximately 0.790, which means that about 79% of the sampled adults reported a willingness to alter their shopping habits if gas prices remain high.
Test Statistic
In hypothesis testing, calculating the test statistic is a critical step. It helps determine whether there is enough evidence against the null hypothesis. The test statistic for proportions often follows a normal distribution, allowing statisticians to use the Z-test. To calculate the Z-test statistic, apply the formula:
  • \(\hat{p}\): The sample proportion
  • \(p_0\): The population proportion under the null hypothesis
  • n: The sample size
For this survey, we aim to test if the proportion of adults willing to alter their shopping habits is greater than 0.75 (75%). Therefore, using the formula,\[ Z = \frac{(\hat{p}-p_0)}{\sqrt{(p_0(1-p_0))/n}} \]Plug in the known values:
  • \(\hat{p} = 0.790\)
  • \(p_0 = 0.75\)
  • n = 1,813
Calculate the Z-value to see how far \(\hat{p}\) is from \(p_0\), adjusted for sample size. This Z-value will guide the decision to accept or reject the null hypothesis.
Null Hypothesis
The null hypothesis (\(H_0\)) plays a pivotal role in hypothesis testing. It is a statement that there is no effect or no difference, serving as a starting point for testing statistical claims. In this survey, the null hypothesis sets the assumption that no more than three-quarters (75%) of the population would change their shopping habits if gas prices remain high. Thus, the statement is formulated as:\[ H_0: p \leq 0.75 \]The researcher seeks evidence for an alternative鈥攖hat more than three-quarters would change their shopping habits:\[ H_1: p > 0.75 \]The goal is to determine whether the survey data provides sufficient evidence to reject the null hypothesis in favor of the alternative. This involves calculating the Z statistic and comparing it to a critical value from the Z distribution, which tells us whether the observed data is statistically significant under the null hypothesis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Explain why you would not reject the null hypothesis if the \(P\) -value were 0.37 .

The paper referenced in the previous exercise also reported that when each of the 1,178 students who participated in the study was asked if he or she played video games at least once a day, 271 responded yes. The researchers were interested in using this information to decide if there is convincing evidence that more than \(20 \%\) of students ages 8 to 18 play video games at least once a day.

The article "The Benefits of Facebook Friends: Social Capital and College Students' Use of Online Social Network Sites" (Journal of Computer-Mediated Communication [2007]: \(1143-1168\) ) describes a study of \(n=286\) undergraduate students at Michigan State University. Suppose that it is reasonable to regard this sample as a random sample of undergraduates at Michigan State. You want to use the survey data to decide if there is evidence that more than \(75 \%\) of the students at this university have a Facebook page that includes a photo of themselves. Let \(p\) denote the proportion of all Michigan State undergraduates who have such a page. (Hint: See Example 10.10\()\) a. Describe the shape, center, and spread of the sampling distribution of \(\hat{p}\) for random samples of size 286 if the null hypothesis \(H_{0}: p=0.75\) is true. b. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.83\) for a sample of size 286 if the null hypothesis \(H_{0}: p=0.75\) were true? Explain why or why not. c. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.79\) for a sample of size 286 if the null hypothesis \(H_{0}: p=0.75\) were true? Explain why or why not. d. The actual sample proportion observed in the study was \(\hat{p}=0.80 .\) Based on this sample proportion, is there convincing evidence that the null hypothesis \(H_{0}: p=\) 0.75 is not true, or is \(\hat{p}\) consistent with what you would expect to see when the null hypothesis is true? Support your answer with a probability calculation.

According to a Washington Post-ABC News poll, 331 of 502 randomly selected U.S. adults said they would not be bothered if the National Security Agency collected records of personal telephone calls. Is there sufficient evidence to conclude that a majority of U.S. adults feel this way? Test the appropriate hypotheses using a 0.01 significance level.

In a hypothesis test, what does it mean to say that the null hypothesis was rejected?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.