/*! 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 29 In a survey of 1000 women age 22... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 29

In a survey of 1000 women age 22 to 35 who work full time, 540 indicated that they would be willing to give up some personal time in order to make more money (USA Today. March 4, 2010 ). The sample was selected in a way that was designed to produce a sample that was representative of women in the targeted age group. a. Do the sample data provide convincing evidence that the majority of women age 22 to 35 who work full-time would be willing to give up some personal time for more money? Test the relevant hypotheses using \(\alpha=.01\). b. Would it be reasonable to generalize the conclusion from Part (a) to all working women? Explain why or why not.

Short Answer

Expert verified
a) If our calculated Z-value from Step 2 is larger than the critical value of 2.33, we can conclude that the majority of women age 22 to 35 who work full-time would be willing to give up some personal time for more money. b) No, the conclusion cannot be generalized to all working women due to the specific age and work status of women in the study.

Step by step solution

01

Set up the hypotheses

We can express our hypotheses as follows: Null hypothesis (H0): p = 0.5 - The proportion of women who would give up personal time is the same as those who would not. Alternative hypothesis (Ha): p > 0.5 - The proportion of women who would give up personal time is more than those who would not.
02

Calculate the test statistic

The Z-value can be calculated using the following formula: Z = \(( \hat{p} - p0) / \sqrt{ \frac{p0*(1-p0)}{n}} \) Where, \hat{p} = proportion in the sample = 540/1000 = 0.54 p0 = assumed population proportion under H0 = 0.5 n = sample size = 1000 The Z-value is calculated as: Z = \( (0.54 - 0.5) / \sqrt{(0.5*(1-0.5))/1000} \)
03

Find the critical value for \(\alpha =.01\)

The critical value for a one-tailed test at \(\alpha = .01\) is Z = 2.33. If our calculated Z-value is greater than this, we reject the null hypothesis.
04

Compare the test statistic to the critical value and make a decision

If the calculated Z-value from Step 2 is larger than 2.33, we reject the null hypothesis and conclude that the majority of women age 22 to 35 who work full-time would give up some personal time for more money.
05

Answer Question b

If the sample was truly representative of the population, we can infer the same for the entire population. However, we cannot generalize this conclusion to all working women, as the sample was specific to women between ages 22 and 35 who work full-time.

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
In hypothesis testing, the null hypothesis is a foundational concept. It is often denoted as \( H_0 \). This hypothesis represents a statement of no effect or no difference. In other words, it suggests that any observed difference in a sample is due to random chance. For the problem at hand, the null hypothesis is that the proportion of women willing to give up personal time for money is the same as those who aren't. Formally, it's expressed as \( p = 0.5 \). This means that 50% of women would and 50% wouldn't give up personal time.
Alternative Hypothesis
Contrary to the null hypothesis, the alternative hypothesis is denoted by \( H_a \). It reflects the researcher's claim or what they aim to prove. Here, it suggests a significant effect or difference, generally contradicting the null hypothesis. For our exercise, the alternative hypothesis states that more than half of women in the specified group are willing to sacrifice personal time for extra money, not just an even split. Mathematically, this means \( p > 0.5 \). If evidence supports this hypothesis, then the claim of majority holds true.
Z-value Calculation
The Z-value, also called the Z-score, is a statistical measure that shows how far away a data point is from the mean in terms of standard deviations. It's critical in determining whether to reject the null hypothesis. In the formula \( Z = \frac{(\hat{p} - p_0)}{\sqrt{\frac{p_0 (1-p_0)}{n}}} \):
  • \( \hat{p} \) is the sample proportion.
  • \( p_0 \) is the assumed population proportion under the null hypothesis.
  • \( n \) refers to the sample size.
In this problem, while \( \hat{p} = 0.54 \) and \( p_0 = 0.5 \), with \( n \) of 1000, this calculation helps us decide whether the evidence strongly contradicts \( H_0 \). If the Z-value is greater than the critical value of 2.33 (for a one-tailed test with \( \alpha = 0.01 \)), we reject \( H_0 \).
Population Proportion
The population proportion \( p \) represents the fraction of the total population that demonstrates a particular attribute. In hypothesis testing, it's an assumed value based on prior knowledge or expectations, under the null hypothesis. In this exercise, the population proportion under \( H_0 \) is set at 0.5. This proportion serves as a benchmark to test against our observed data from the sample. Understanding this concept helps frame our expectations and guides the null hypothesis formation.
Sample Size
Sample size, designated as \( n \), is a crucial part of hypothesis testing. It refers to the number of observations or data points collected from the population. A sufficiently large sample size is important to ensure the results are reliable and not due to random variation. In our exercise, the sample size is 1000, indicating a substantial group, giving weight to the findings and making them more credible. The larger the sample size, the more likely it is to accurately represent the larger population, lending confidence to the conclusions drawn from the test.

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

"Most Like it Hot" is the title of a press release issued by the Pew Research Center (March 18, 2009 , www.pewsocialtrends.org). The press release states that "by an overwhelming margin, Americans want to live in a sunny place." This statement is based on data from a nationally representative sample of 2260 adult Americans. Of those surveyed, 1288 indicated that they would prefer to live in a hot climate rather than a cold climate. Do the sample data provide convincing evidence that a majority of all adult Americans prefer a hot climate over a cold climate? Use the nine-step hypothesis testing process with \(\alpha=.01\) to answer this question.

Paint used to paint lines on roads must reflect enough light to be clearly visible at night. Let \(\mu\) denote the mean reflectometer reading for a new type of paint under consideration. A test of \(H_{0}: \mu=20\) versus \(H_{a}: \mu>20\) based on a sample of 15 observations gave \(t=3.2\). What conclusion is appropriate at each of the following significance levels? a. \(\quad \alpha=.05\) c. \(\alpha=.001\) b. \(\quad \alpha=.01\)

To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose that the specifications state that the mean strength of welds should exceed \(100 \mathrm{lb} / \mathrm{in}^{2}\). The inspection team decides to test \(H_{0}: \mu=100\) versus \(H_{a}: \mu>100 .\) Explain why this alternative hypothesis was chosen rather than \(\mu<100\).

Many consumers pay careful attention to stated nutritional contents on packaged foods when making purchases. It is therefore important that the information on packages be accurate. A random sample of \(n=12\) frozen dinners of a certain type was selected from production during a particular period, and the calorie content of each one was determined. (This determination entails destroying the product, so a census would certainly not be desirable!) Here are the resulting observations, along with a boxplot and normal probability plot: \(\begin{array}{llllllll}255 & 244 & 239 & 242 & 265 & 245 & 259 & 248\end{array}\) \(\begin{array}{llll}225 & 226 & 251 & 233\end{array}\) a. Is it reasonable to test hypotheses about mean calorie content \(\mu\) by using a \(t\) test? Explain why or why not. b. The stated calorie content is \(240 .\) Does the boxplot suggest that true average content differs from the stated value? Explain your reasoning. c. Carry out a formal test of the hypotheses suggested in Part (b).

Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40 -amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact \(40 .\) If the mean amperage is lower than \(40,\) customers will complain because the fuses require replacement too often. If the mean amperage is higher than \(40,\) the manufacturer might be liable for damage to an electrical system as a result of fuse malfunction. To verify the mean amperage of the fuses, a sample of fuses is selected and tested. If a hypothesis test is performed using the resulting data, what null and alternative hypotheses would be of interest to the manufacturer?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

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.