/*! 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 58 What is the shape of the samplin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 58

What is the shape of the sampling distribution of \(\hat{p}_{1}-\hat{p}_{2}\) for two large samples? What are the mean and standard deviation of this sampling distribution?

Short Answer

Expert verified
The shape of the sampling distribution of \(\hat{p}_{1}-\hat{p}_{2}\) for two large samples is a normal distribution. The mean of the sampling distribution is \(p_{1} - p_{2}\), which is the difference between the population proportions. The standard deviation can be obtained via \(\sqrt{\frac{{p_{1}(1-p_{1})}}{{n_{1}}}+ \frac{{p_{2}(1-p_{2})}}{{n_{2}}}}\).

Step by step solution

01

Shape of the sampling distribution

Under the condition that the sample sizes are large, the Central Limit Theorem assures that the distribution of \(\hat{p}_{1}-\hat{p}_{2}\) will be a normal distribution.
02

Calculation of the Mean

The mean (expected value) of the sampling distribution \(\hat{p}_{1}-\hat{p}_{2}\) is equal to the difference between the population proportions, \(p_{1} - p_{2}\).
03

Calculation of the Standard Deviation

The standard deviation of the sampling distribution \(\hat{p}_{1}-\hat{p}_{2}\) can be calculated using the formula: \(\sqrt{\frac{{p_{1}(1-p_{1})}}{{n_{1}}}+ \frac{{p_{2}(1-p_{2})}}{{n_{2}}}}\), where \(p_{1}\) and \(p_{2}\) are the population proportions, and \(n_{1}\) and \(n_{2}\) are the sizes of the samples respectively.

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!

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

Two competing airlines, Alpha and Beta, fly a route between Des Moines, Iowa, and Wichita, Kansas. Each airline claims to have a lower percentage of flights that arrive late. Let \(p_{1}\) be the proportion of Alpha's flights that arrive late and \(p_{2}\) the proportion of Beta's tlights that arrive late. a. You are asked to observe a random sample of arrivals for each airline to estimate \(p_{1}-p_{2}\) with a \(90 \%\) confidence level and a margin of error of estimate of \(.05 .\) How many arrivals for each airline would you have to observe? (Assume that you will observe the same number of arrivals, \(n\), for each airline. To be sure of taking a large enough sample, use \(p_{1}=p_{2}=.50\) in your calculations for \(n .\) ) b. Suppose that \(p_{1}\) is actually \(.30\) and \(p_{2}\) is actually \(.23 .\) What is the probability that a sample of 100 flights for each airline ( 200 in all) would yield \(\hat{p}_{1} \geq \hat{p}_{2}\) ?

According to an estimate, the average earnings of female workers who are not union members are \(\$ 388\) per week and those of female workers who are union members are \(\$ 505\) per week. Suppose that these average earnings are calculated based on random samples of 1500 female workers who are not union members and 2000 female workers who are union members. Further assume that the standard deviations for the two corresponding populations are \(\$ 30\) and \(\$ 35\), respectively. a. Construct a \(95 \%\) confidence interval for the difference between the two population means. b. Test at the \(2.5 \%\) significance level whether the mean weekly earnings of female workers who are not union members are less than those of female workers who are union members.

Conduct the following tests of hypotheses, assuming that the populations of paired differences are normally distributed a. \(H_{0}: \mu_{d}=0, \quad H_{1}: \mu_{d} \neq 0, \quad n=26, \quad \bar{d}=9.6, \quad s_{d}=3.9, \quad \alpha=.05\) b. \(H_{0}: \mu_{d}=0, \quad H_{1}: \mu_{d}>0, \quad n=15, \quad \bar{d}=8.8, \quad s_{d}=4.7, \quad \alpha=.01\) c. \(H_{0}: \mu_{d}=0, \quad H_{1}: \mu_{d}<0, \quad n=20, \quad \bar{d}=-7.4, \quad s_{d}=2.3, \quad \alpha=.10\)

The following information was obtained from two independent samples selected from two populations with unknown but equal standard deviations. $$ \begin{array}{lll} n_{1}=55 & \bar{x}_{1}=90.40 & s_{1}=11.60 \\ n_{2}=50 & \bar{x}_{2}=86.30 & s_{2}=10.25 \end{array} $$ a. What is the point estimate of \(\mu_{1}-\mu_{2}\) ? b. Construct a \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\).

We wish to estimate the difference between the mean scores on a standardized test of students taught by Instructors \(\mathrm{A}\) and \(\mathrm{B}\). The scores of all students taught by Instructor A have a normal distribution with a standard deviation of 15, and the scores of all students taught by Instructor B have a normal distribution with a standard deviation of 10 . To estimate the difference between the two means, you decide that the same number of students from each instructor's class should be observed. a. Assuming that the sample size is the same for each instructor's class, how large a sample should be taken from each class to estimate the difference between the mean scores of the two populations to within 5 points with \(90 \%\) confidence? b. Suppose that samples of the size computed in part a will be selected in order to test for the difference between the two population mean scores using a \(.05\) level of significance. How large does the difference between the two sample means have to be for you to conclude that the two population means are different? c. Explain why a paired-samples design would be inappropriate for comparing the scores of Instructor A versus Instructor \(\mathrm{B}\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.