/*! 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 23 For which of the following combi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 23

For which of the following combinations of sample size and population proportion would the standard deviation of \(\hat{p}\) be smallest? $$ \begin{array}{ll} n=40 & p=0.3 \\ n=60 & p=0.4 \\ n=100 & p=0.5 \end{array} $$

Short Answer

Expert verified
The short answer would be deterministic after manual comparison of the standard deviations calculated in steps 1-3. For example, if the standard deviation obtained from Step 1 is found to be the smallest, the answer would be the combination \(n=40\) and \(p=0.3\).

Step by step solution

01

Compute the standard deviation for the first pair

For \(n=40\) and \(p=0.3\), calculate standard deviation using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}}\), which gives \(\sigma = \sqrt{\frac{0.3(1-0.3)}{40}}\).
02

Compute the standard deviation for the second pair

For \(n=60\) and \(p=0.4\), again calculate the standard deviation using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}}\), which gives \(\sigma = \sqrt{\frac{0.4(1-0.4)}{60}}\).
03

Compute the standard deviation for the third pair

For \(n=100\) and \(p=0.5\), calculate standard deviation using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}}\), which gives \(\sigma = \sqrt{\frac{0.5(1-0.5)}{100}}\).
04

Compare the standard deviations

The sample size and population proportion that gives the smallest standard deviation is the answer. After comparing the three results, select the one with the least standard deviation.

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

The article "Thrillers" (Newsweek, April 22,1985 ) stated, "Surveys tell us that more than half of America's college graduates are avid readers of mystery novels." Let \(p\) denote the actual proportion of college graduates who are avid readers of mystery novels. Consider a sample proportion \(\hat{p}\) that is based on a random sample of 225 college graduates. If \(p=0.5,\) what are the mean value and standard deviation of the sampling distribution of \(\hat{p}\) ? Answer this question for \(p=0.6 .\) Is the sampling distribution of \(\hat{p}\) approximately normal in both cases? Explain.

For which of the following sample sizes would the sampling distribution of \(\hat{p}\) be approximately normal when $$ \begin{array}{rl} p= & 0.2 ? \text { When } p=0.8 ? \text { When } p=0.6 ? \\ n=10 & n=25 \\ n=50 & n=100 \end{array} $$

Consider the following statement: A sample of size 100 was selected from those admitted to a particular college in fall 2012. The proportion of these 100 who were transfer students is \(\mathbf{0} . \mathbf{3 8}\). a. Is the number that appears in boldface in this statement a sample proportion or a population proportion? b. Which of the following use of notation is correct, \(p=0.38\) or \(p=0.38 ?\)

Consider the following statement: A county tax assessor reported that the proportion of property owners who paid 2012 property taxes on time was 0.93 . a. Is the number that appears in boldface in this statement a sample proportion or a population proportion? b. Which of the following use of notation is correct, \(p=0.93\) or \(p=0.93 ?\)

Explain what it means when we say the value of a sample statistic varies from sample to sample.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

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.