/*! 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 2 CareerBuilder.com conducted a su... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 2

CareerBuilder.com conducted a survey to learn about the proportion of employers who had ever sent an employee home because they were dressed inappropriately (June \(17,2008,\) www. careerbuilder.com). Suppose you are interested in determining if the resulting data provide strong evidence in support of the claim that more than one-third of employers have sent an employee home to change clothes. To answer this question, what null and alternative hypotheses should you test?

Short Answer

Expert verified
The null hypothesis is \(H_0: p = 1/3\) and the alternative hypothesis is \(H_a: p > 1/3\)

Step by step solution

01

Identify the Null Hypothesis

The null hypothesis is a proposition that there is no effect or difference in the population. This is represented as \(H_0\). In this situation, since the question is concerning the claim that more than a third of employers send an employee home to change clothes, the null hypothesis would be that a third or less employers send an employee home to change clothes. Therefore, \(H_0: p = 1/3\), where \(p\) is the proportion of employers who have sent an employee home due to inappropriate dressing.
02

Identify the Alternative Hypothesis

The alternative hypothesis is a proposition that there is a difference or effect in the population and contrasts with the null hypothesis. This is represented as \(H_a\). Given the claim that more than a third of employers send an employee home to change clothes, the alternative hypothesis would be that more than a third of employers send an employee home to change clothes. Therefore, \(H_a: p > 1/3\)

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.

Null Hypothesis
The Null Hypothesis is an essential concept in hypothesis testing. It's like the default assumption or the starting point. When we perform a hypothesis test, we initially assume that the null hypothesis is true. This is a statement of no effect or no difference. Imagine you are investigating whether a new teaching method improves test scores. The null hypothesis would claim that the new method has no effect on scores compared to the traditional one.In the context of our example with CareerBuilder.com, the null hypothesis suggests that the proportion of employers who have sent an employee home for dressing inappropriately is one-third or less.- The mathematical notation for this is: - \(H_0: p \leq \frac{1}{3}\)- Here, \(p\) signifies the true proportion of employers.Remember, the purpose of testing the null hypothesis is to challenge it with data. But unless we find strong evidence against it, we maintain this assumption.
Alternative Hypothesis
The Alternative Hypothesis stands in opposition to the null. It's the statement suggesting there's an effect, a difference, or a change. When our data provides sufficient evidence in its favor, we reject the null hypothesis.In our CareerBuilder.com example, the alternative hypothesis posits that more than one-third of employers have sent home an employee due to inappropriate attire. It's what we seek to prove with our data:- The mathematical notation is: - \(H_a: p > \frac{1}{3}\)- Here, \(p\) denotes the actual proportion we're investigating.When conducting a test, we're ultimately trying to see if there's enough data to support the alternative hypothesis. It's like determining if there are sufficient grounds to believe the claim being tested.
Proportion Hypothesis Test
A Proportion Hypothesis Test allows us to evaluate claims concerning proportions in a population based on sample data. This statistical method helps in making decisions about the population proportion \(p\).To perform the test, follow these steps:
  • Define the null and alternative hypotheses. In our example, these are \(H_0: p = \frac{1}{3}\) and \(H_a: p > \frac{1}{3}\).
  • Collect sample data and calculate the sample proportion \(\hat{p}\).
  • Determine the standard error using the formula: \( SE = \sqrt{\frac{p(1-p)}{n}} \), where \(n\) is the sample size.
  • Find the test statistic (z-score): \(z = \frac{\hat{p} - p}{SE}\).
  • Compare the test statistic to a critical value from the z-distribution to decide whether to reject the null hypothesis.
This structured approach makes it feasible to make informed decisions about the underlying claims. Proportion hypothesis tests are invaluable in fields ranging from social science to marketing research, where understanding population dynamics through sample insights is critical.

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

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.

The report "How Teens Use Media" (Nielsen, June 2009) says that \(37 \%\) of U.S. teens access the Internet from a mobile phone. Suppose you plan to select a random sample of students at the local high school and will ask each student in the sample if he or she accesses the Internet from a mobile phone. You want to determine if there is evidence that the proportion of students at the high school who access the Internet using a mobile phone differs from the national figure of 0.37 given in the Nielsen report. What hypotheses should you test?

A college has decided to introduce the use of plus and minus with letter grades, as long as there is convincing evidence that more than \(60 \%\) of the faculty favor the change. A random sample of faculty will be selected, and the resulting data will be used to test the relevant hypotheses. If \(p\) represents the proportion of all faculty who favor a change to plus-minus grading, which of the following pairs of hypotheses should be tested? $$H_{0}: p=0.6 \text { versus } H_{a}: p<0.6$$ or $$H_{0}: p=0.6 \text { versus } H_{a}: p>0.6$$ Explain your choice.

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes is the large-sample \(z\) test appropriate? a. \(H_{0}: p=0.8, n=40\) b. \(H_{0}: p=0.4, n=100\) c. \(H_{0}: p=0.1, n=50\) d. \(H_{0}: p=0.05, n=750\)

Every year on Groundhog Day (February 2), the famous groundhog Punxsutawney Phil tries to predict whether there will be 6 more weeks of winter. The article "Groundhog Has Been Off Target" (USA Today, Feb. 1,2011 ) states that "based on weather data, there is no predictive skill for the groundhog." Suppose that you plan to take a random sample of 20 years and use weather data to determine the proportion of these years the groundhog's prediction was correct. a. Describe the shape, center, and spread of the sampling distribution of \(\hat{p}\) for samples of size 20 if the groundhog has only a \(50-50\) chance of making a correct prediction. b. Based on your answer to Part (a), what sample proportion values would convince you that the groundhog's predictions have a better than \(50-50\) chance of being correct?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.