/*! 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 The article "Thrillers" (Newswee... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 29

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 \(\pi\) denote the actual proportion of college graduates who are avid readers of mystery novels. Consider a sample proportion \(p\) that is based on a random sample of 225 college graduates. a. If \(\pi=.5\), what are the mean value and standard deviation of \(p ?\) Answer this question for \(\pi=.6\). Does \(p\) have approximately a normal distribution in both cases? Explain. b. Calculate \(P(p \geq .6)\) for both \(\pi=.5\) and \(\pi=.6\). c. Without doing any calculations, how do you think the hange if \(n\) were 400 rather

Short Answer

Expert verified
a) For \(\pi = 0.5\), Mean value = \(0.5\) and Standard deviation = \(0.033\); For \(\pi = 0.6\), Mean value = \(0.6\) and Standard deviation = \(0.032\). The distribution is approximately normal in both cases as sample size \(n = 225\) which is sufficiently large. b) For \(\pi = 0.5\), Z = 3.03; and for \(\pi = 0.6\), Z = 0. Look up these Z-scores in a standard normal distribution table to find corresponding probabilities. c) Increasing sample size to 400 from 225 would refine the estimates and make the distribution narrower and more normal due to large sample size.

Step by step solution

01

Calculating the Mean and Standard Deviation

Given \(\pi\) (actual proportion) and \(n\) (sample size), the mean (\(渭\)) of the sample proportion \(p\) is equal to \(\pi\), and the standard deviation (\(蟽\)) is \(\sqrt{(\pi(1-\pi))/n}\). So, for \(\pi = 0.5\) and \(n = 225\), the calculations would be: \[渭 = \pi = 0.5 \ 蟽 = \sqrt{(0.5(1-0.5))/225} = 0.033\] Similarly, for \(\pi = 0.6\), the calculations would be: \[渭 = \pi = 0.6 \ 蟽 = \sqrt{(0.6(1-0.6))/225} = 0.032\]
02

Determining If \(p\) Follows a Normal Distribution

According to the Central Limit Theorem, if a sample size is large (usually if \(n > 30\)) and the samples are drawn independently, the sampling distribution of the mean approaches a normal distribution, regardless of the shape of the population distribution. Here, since \(n = 225\), which is significantly larger than 30, we can approximate the distribution of \(p\) as being approximately normal in both cases.
03

Calculating \(P(p 鈮� .6)\) For Both \(\pi = .5\) and \(\pi = .6\)

This is finding the probability that the proportion of readers is at least 0.6. To do this for \(\pi = 0.5\), use the standard normal (Z) distribution where \(Z = (p-渭)/蟽\), so compute Z when \(p = 0.6\). Using the calculated \(渭\) and \(蟽\) from previous step: \[Z = (0.6 - 0.5)/0.033 = 3.03\] This Z-score is used to find probability from standard normal distribution tables. Similarly, compute Z for \(\pi = 0.6\): \[Z = (0.6 - 0.6)/0.032 = 0\] Use these Z-scores to find probabilities using the standard normal distribution table.
04

Discussing Sample Size Effect

Without doing the actual calculations, it is known in statistics that as sample size \(n\) increases, the distribution of the sample proportion \(p\) tends towards a normal distribution due to the Central Limit Theorem. Also, the standard deviation gets smaller which makes the distribution narrower. Therefore, increasing the sample size from 225 to 400 would make the distribution more normal and increase the precision of the estimates.

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.

Central Limit Theorem
One of the most powerful principles in statistics is the Central Limit Theorem (CLT). This theorem is the reason researchers can make inferences about population parameters even when they only have access to a small part of the population.

CLT states that, given a sufficiently large sample size, the sampling distribution of the mean for a random variable will approximate a normal distribution, regardless of the distribution of the original variable. In simple terms, if you keep taking samples and calculating their means, the graph of these means will form a bell-shaped curve known as a normal distribution.

Here's the kicker: this magic starts happening with sample sizes as small as 30 data points! But remember, the larger the sample, the better the approximation to a normal distribution will be. This holds true for sample proportions as well, which relates directly to our Newsweek exercise question about avid readers of mystery novels among college graduates.
Normal Distribution
Imagine a perfectly symmetrical bell鈥攖hat's a normal distribution. It's a key concept in statistics, representing a continuous probability distribution where the majority of observations cluster around the mean, and there are fewer and fewer observations as you move away from the center. The points drop off symmetrically, creating what statisticians call the 'bell curve.'

In the context of our discussion, the sample proportion (avid mystery novel readers) will tend to follow a normal distribution as the sample size increases, thanks to the CLT. This makes predicting outcomes and drawing conclusions much more straightforward. For instance, given a large enough group of college graduates, we can predict that the number of mystery novel enthusiasts will likely fall in a particular range, with a predictable variation.
Sample Proportion
In statistics, a sample proportion is just what it sounds like鈥攊t's the proportion of a certain outcome within a sample. It's often used as an estimate for the proportion in the entire population. For our thrills and mystery-interest example, it's the percentage of college graduates in our sample who love a good 'whodunit' novel.

Calculating this proportion is straightforward: if you have 50 avid mystery-reading graduates out of a sample of 200, your sample proportion is 50/200, or 0.25 (25%). But, there's more! This proportion has its own mean (equal to the true proportion in the whole population) and standard deviation, which can be used to determine the likely variation from the mean. These properties enable us to predict the range in which the actual population proportion may lie, with a given level of confidence.

In the Newsweek exercise, the sample mean for the proportion is simply the stated population proportion, while its standard deviation reflects how much the sample proportion is expected to vary from the population proportion due to random chance.

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

Suppose that \(20 \%\) of the subscribers of a cable television company watch the shopping channel at least once a week. The cable company is trying to decide whether to replace this channel with a new local station. A survey of 100 subscribers will be undertaken. The cable company has decided to keep the shopping channel if the sample proportion is greater than \(.25\). What is the approximate probability that the cable company will keep the shopping channel, even though the true proportion who watch it is only .20?

A random sample is to be selected from a population that has a proportion of successes \(\pi=.65 .\) Determine the mean and standard deviation of the sampling distribution of \(p\) for each of the following sample sizes: a. \(n=10\) b. \(n=20\) c. \(n=30\) d. \(n=50\) e. \(n=100\) f. \(n=200\)

In the library on a university campus, there is a sign in the elevator that indicates a limit of 16 persons. Furthermore, there is a weight limit of \(2500 \mathrm{lb}\). Assume that the average weight of students, faculty, and staff on campus is \(150 \mathrm{lb}\), that the standard deviation is \(27 \mathrm{lb}\), and that the distribution of weights of individuals on campus is approximately normal. If a random sample of 16 persons from the campus is to be taken: a. What is the mean value of the distribution of the sample mean? b. What is the standard deviation of the sampling distribution of the sample mean weight? c. What average weights for a sample of 16 people will result in the total weight exceeding the weight limit of \(2500 \mathrm{lb}\) ? d. What is the chance that a random sample of 16 persons on the elevator will exceed the weight limit?

A certain chromosome defect occurs in only 1 out of 200 adult Caucasian males. A random sample of \(n=100\) adult Caucasian males is to be obtained. a. What is the mean value of the sample proportion \(p\), and what is the standard deviation of the sample proportion? b. Does \(p\) have approximately a normal distribution in this case? Explain. c. What is the smallest value of \(n\) for which the sampling distribution of \(p\) is approximately normal?

For each of the following statements, identify the number that appears in boldface type as the value of either a population characteristic or a statistic: a. A department store reports that \(84 \%\) of all customers who use the store's credit plan pay their bills on time. b. A sample of 100 students at a large university had a mean age of \(24.1\) years. c. The Department of Motor Vehicles reports that \(22 \%\) of all vehicles registered in a particular state are imports. d. A hospital reports that based on the 10 most recent cases, the mean length of stay for surgical patients is 6.4 days. e. A consumer group, after testing 100 batteries of a certain brand, reported an average life of \(63 \mathrm{hr}\) of use.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.