/*! 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 8 A study is to be performed to es... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 8

A study is to be performed to estimate the proportion of voters who believe the economy is "heading in the right direction." Which of the following pairs of sample size and population proportion \(p\) will result in the smallest variance for the sampling distribution of \(\hat{p} ?\) (A) \(n=100\) and \(p=0.1\) (B) \(n=100\) and \(p=0.5\) (C) \(n=100\) and \(p=0.99\) (D) \(n=1000\) and \(p=0.1\) (E) \(n=1000\) and \(p=0.5\)

Short Answer

Expert verified
Pair (D) with \(n = 1000\) and \(p = 0.1\).

Step by step solution

01

Understand the variance formula for sampling distribution

The variance for the sampling distribution of the sample proportion \(\hat{p}\) is given by \[\text{Var}(\hat{p}) = \frac{p(1 - p)}{n}\]. We need to calculate this variance for each pair \(n\) and \(p\).
02

Calculate variance for (A)

For \(n = 100\) and \(p = 0.1\), \[\text{Var}(\hat{p}) = \frac{0.1(1 - 0.1)}{100} = \frac{0.1 \times 0.9}{100} = \frac{0.09}{100} = 0.0009\]
03

Calculate variance for (B)

For \(n = 100\) and \(p = 0.5\), \[\text{Var}(\hat{p}) = \frac{0.5(1 - 0.5)}{100} = \frac{0.5 \times 0.5}{100} = \frac{0.25}{100} = 0.0025\]
04

Calculate variance for (C)

For \(n = 100\) and \(p = 0.99\), \[\text{Var}(\hat{p}) = \frac{0.99(1 - 0.99)}{100} = \frac{0.99 \times 0.01}{100} = \frac{0.0099}{100} = 0.000099\]
05

Calculate variance for (D)

For \(n = 1000\) and \(p = 0.1\), \[\text{Var}(\hat{p}) = \frac{0.1(1 - 0.1)}{1000} = \frac{0.1 \times 0.9}{1000} = \frac{0.09}{1000} = 0.00009\]
06

Calculate variance for (E)

For \(n = 1000\) and \(p = 0.5\), \[\text{Var}(\hat{p}) = \frac{0.5(1 - 0.5)}{1000} = \frac{0.5 \times 0.5}{1000} = \frac{0.25}{1000} = 0.00025\]
07

Compare the variances

The calculated variances are as follows: \((A) = 0.0009\), \((B) = 0.0025\), \((C) = 0.000099\), \((D) = 0.00009\), \((E) = 0.00025\). The smallest variance is 0.00009 from pair (D).

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.

sampling distribution
When studying statistics, you often work with sampling distributions. A sampling distribution captures different outcomes for a sample statistic, such as the sample proportion \(\hat{p}\), based on different random samples from a population. It helps us understand how a sample proportion might vary from one sample to another. Imagine performing many repeated random samples from a population; the sampling distribution of \(\hat{p}\) shows us the distribution of those sample proportions. This concept is vital because it allows us to make predictions about the population based on our sample data.
variance formula
The variance of the sampling distribution of the sample proportion \(\hat{p}\) is a measure of how much the proportion values differ across various samples. This variance is given by the formula: \[ \text{Var}(\hat{p}) = \frac{p(1 - p)}{n} \]where \(p\) is the population proportion and \(n\) is the sample size.
The formula tells us that higher sample sizes (larger \(n\)) yield smaller variances, making our sample proportion estimates more precise. Furthermore, variance also depends on the proportion \(p\). Notice how the product \(p(1 - p)\) varies. When \(p\) is close to 0.5, it鈥檚 maximized, creating a larger variance.
sample proportion
The sample proportion \(\hat{p}\) is an estimate based on observed data. For example, if you're surveying people's opinions about the economy鈥檚 direction, \(\hat{p}\) would represent the proportion of surveyed voters who believe the economy is heading in the right direction. To calculate \(\hat{p}\), you use the formula:
\[ \hat{p} = \frac{X}{n} \]
Here, \(X\) is the number of successes (e.g., voters who believe the economy is heading in the right direction), and \(n\) is the total number of observations in your sample. This value gives you a point estimate of the population proportion \(p\).
population proportion
The population proportion \(p\) represents the true proportion of the population that possesses a certain characteristic 鈥� such as the percentage of all voters believing the economy is heading in the right direction. Unlike the sample proportion \(\hat{p}\), \(p\) is a fixed value, though unknown. To estimate \(p\), we rely on \(\hat{p}\), the sample proportion. By using a formula that adjusts the sample proportion based on the size of the sample and variability within the population, we get a more accurate picture of \(p\). Understanding \(p\) helps in making informed decisions or predictions about future samples or other population characteristics.

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

Which of the following are unbiased estimators for the corresponding population parameters? I. Sample means II. Sample proportions III. Difference of sample means IV. Difference of sample proportions (A) None are unbiased. (B) I and II only (C) I and III only (D) III and IV only (E) All are unbiased.

Assume the given distributions are roughly normal. A coffee machine can be adjusted to deliver any fixed number of ounces of coffee. If the machine has a standard deviation in delivery equal to 0.4 ounce, what should be the mean setting so that an 8-ounce cup will overflow only \(0.5 \%\) of the time? (A) \(8-0.995(0.4)\) (B) \(8+1.96(0.4)\) (C) \(8-1.96(0.4)\) (D) \(8+2.576(0.4)\) (E) \(8-2.576(0.4)\)

Which of the following is an incorrect statement? (A) The sampling distribution of \(\bar{x}\) has mean equal to the population mean \(\mu\) even if the population is not normally distributed. (B) The sampling distribution of \(\bar{x}\) has standard deviation \(\frac{\sigma}{\sqrt{n}}\) even if the population is not normally distributed. (C) The sampling distribution of \(\bar{x}\) is normal if the population has a normal distribution. (D) When \(n\) is large, the sampling distribution of \(\bar{x}\) is approximately normal even if the population is not normally distributed. (E) The larger the value of the sample size \(n\), the closer the standard deviation of the sampling distribution of \(\bar{x}\) is to the standard deviation of the population.

Suppose a population is skewed right. For which of the following sample sizes would the sampling distribution of \(\bar{x}\) be closest to normal? (A) 10 (B) 30 (C) 50 (D) 100 (E) According to the central limit theorem, all give normal sampling distributions.

Researchers found that in the aftermath of the 2011 Fukushima nuclear disaster, \(12 \%\) of the pale grass blue butterfly larvae developed mutations as adults. What is the probability that in a random sample of 300 of these butterfly larvae, more than \(15 \%\) developed mutations as adults? (A) \(P\left(z>\frac{0.12-0.15}{\sqrt{\frac{(0.12)(0.88)}{300}}}\right)\) (B) \(P\left(z>\frac{0.12-0.15}{\sqrt{\frac{(0.12)(0.88)}{300}}}\right)\) (C) \(P\left(z>\frac{0.12-0.15}{\sqrt{\frac{(0.12)(0.88)}{300}}}\right)\) (D) \(P\left(z>\frac{0.15-0.12}{\left(\frac{(0.12)(0.88)}{\sqrt{300}}\right)}\right)\) (E) \(P\left(z>\frac{0.15-0.12}{\left(\frac{(0.12)(0.88)}{\sqrt{300}}\right)}\right)\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.