/*! 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 38 Is it appropriate to use the nor... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 38

Is it appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p}\) in the following circumstances? a. \(n=50, p=.05\) b. \(n=75, p=.1\) c. \(n=250, p=.99\)

Short Answer

Expert verified
a) n = 50, p = 0.05 b) n = 75, p = 0.1 c) n = 250, p = 0.99 **Answer** a) No, it is not appropriate, as the condition \(np \geq 10\) is not met. b) No, it is not appropriate, as the condition \(np \geq 10\) is not met. c) No, it is not appropriate, as the condition \(n(1-p) \geq 10\) is not met.

Step by step solution

01

Part a: n = 50, p = 0.05

First, let's check if the sample size is large enough and if the two conditions are met: \(n = 50\) \(p = 0.05\) Condition 1: \(np = 50 * 0.05 = 2.5\) Condition 2: \(n(1-p) = 50 * (1 - 0.05) = 47.5\) The sampling distribution does not meet the condition \(np \geq 10\). So, it is not appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p}\) for this circumstance.
02

Part b: n = 75, p = 0.1

Let's check if the sample size is large enough and if the two conditions are met: \(n = 75\) \(p = 0.1\) Condition 1: \(np = 75 * 0.1 = 7.5\) Condition 2: \(n(1-p) = 75 * (1 - 0.1) = 67.5\) The sampling distribution does not meet the condition \(np \geq 10\). So, it is not appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p}\) for this circumstance.
03

Part c: n = 250, p = 0.99

Let's check if the sample size is large enough and if the two conditions are met: \(n = 250\) \(p = 0.99\) Condition 1: \(np = 250 * 0.99 = 247.5\) Condition 2: \(n(1-p) = 250 * (1 - 0.99) = 2.5\) The sampling distribution does not meet the condition \(n(1-p) \geq 10\). So, it is not appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p}\) for this circumstance. In conclusion, for all three circumstances provided, it is not appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p}\).

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 battle for consumer preference continues between Pepsi and Coke. How can you make your preferences known? There is a web page where you can vote for one of these colas if you click on the link that says PAY CASH for your opinion. Explain why the respondents do not represent a random sample of the opinions of purchasers or drinkers of these drinks. Explain the types of distortions that could creep into an Internet opinion poll.

Random samples of size \(n=500\) were selected from a binomial population with \(p=.1\). a. Is it appropriate to use the normal distribution to approximate the sampling distribution of \(\hat{p} ?\) Check to make sure the necessary conditions are met. Using the results of part a, find these probabilities: b. \(\hat{p}>.12\) c. \(\hat{p}<.10\) d. \(\hat{p}\) lies within .02 of \(p\)

A study of about \(n=1000\) individuals in the United States during September \(21-22,2001\) revealed that \(43 \%\) of the respondents indicated that they were less willing to fly following the events of September \(11,2001 .^{19}\) a. Is this an observational study or a designed experiment? b. What problems might or could have occurred because of the sensitive nature of the subject? What kinds of biases might have occurred?

Random samples of size \(n\) were selected from populations with the means and variances given here. Find the mean and standard deviation of the sampling distribution of the sample mean in each case: a. \(n=36, \mu=10, \sigma^{2}=9\) b. \(n=100, \mu=5, \sigma^{2}=4\) c. \(n=8, \mu=120, \sigma^{2}=1\)

Suppose a random sample of \(n=5\) observations is selected from a population that is normally distributed, with mean equal to 1 and standard deviation equal to \(.36 .\) a. Give the mean and the standard deviation of the sampling distribution of \(\bar{x}\). b. Find the probability that \(\bar{x}\) exceeds 1.3 , using the Normal Probabilities for Means applet. c. Find the probability that the sample mean \(\bar{x}\) is less than .5. d. Find the probability that the sample mean deviates from the population mean \(\mu=1\) by more than .4 .
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Discrete 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.