/*! 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 71 Consider a large population with... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 71

Consider a large population with \(p=.21 .\) Assuming \(n / N \leq .05\), find the mean and standard deviation of the sample proportion \(\hat{p}\) for a sample size of a. 400 b. 750

Short Answer

Expert verified
The mean of the sample proportion is 0.21 for both samples (400 and 750). The standard deviation of the sample proportion for a sample size of 400 is calculated to be approximately 0.022 and for a sample size of 750 to be approximately 0.017.

Step by step solution

01

Find the mean for the sample sizes

The mean of the sample proportion (also known as the expected value of p-hat) is equal to the population proportion (p). So, the mean is 0.21 for both samples (400 and 750).
02

Find the standard deviation for a sample size of 400

The standard deviation of the sample proportion for a sample size of 400 can be found using the formula \(\sigma=\sqrt{p(1-p)/n}\). Substituting given values we get \(\sigma=\sqrt{0.21*(1-0.21)/400}\).
03

Find the standard deviation for a sample size of 750

Similar to Step 2, to find the standard deviation of the sample proportion for a sample size of 750, use the formula \(\sigma=\sqrt{p(1-p)/n}\). Substituting given values into the formula, \(\sigma=\sqrt{0.21*(1-0.21)/750}\).

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.

Understanding Sample Proportion
When dealing with large populations, we often rely on samples. To represent the proportion of interest in a sample, we use the sample proportion, denoted as \( \hat{p} \). Essentially, \( \hat{p} \) is an estimate of the true population proportion \( p \).
For example, if in a sample of 100 people, 30 have a certain characteristic, the sample proportion \( \hat{p} \) is calculated as 30 divided by 100, which equals 0.30.
Sample proportions are vital in statistical inference because they help us make predictions and form conclusions about the larger population. Remember, while the sample proportion provides a snapshot, it's always just an estimate of the true population proportion.
Mean of Sample Proportion
The mean of the sample proportion (expected value of \( \hat{p} \)) is fundamentally the true proportion of the population, \( p \).
For example, if the true proportion \( p \) is 0.21, then the mean of the sample proportion \( \mu_{\hat{p}} \) is also 0.21. This tells us that, on average, our sample proportions will center around this true proportion if we take many samples.
This equality between the mean of \( \hat{p} \) and \( p \) is a result of choosing samples randomly, ensuring unbiased representations of the population.
  • Example: In the problem, regardless of whether the sample size is 400 or 750, the mean sample proportion remains 0.21, because the population proportion \( p \) is 0.21.
Standard Deviation of Sample Proportion
The standard deviation of the sample proportion \( \hat{p} \) quantifies how much \( \hat{p} \) is expected to vary from sample to sample.
Calculated using the formula \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \), it depends on both the true proportion and the sample size \( n \).
  • In essence, the larger the sample size, the smaller the standard deviation. This means that larger samples produce more accurate estimates of the population proportion.
  • In the original exercise, with \( p = 0.21 \), the standard deviation for a sample size of 400 was calculated as \( \sigma_{\hat{p}} = \sqrt{\frac{0.21(1-0.21)}{400}} \), resulting in a certain value. Similarly, for a sample size of 750, the calculation was \( \sigma_{\hat{p}} = \sqrt{\frac{0.21(1-0.21)}{750}} \), which yields a smaller standard deviation than a sample size of 400, illustrating the concept further.

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

If all possible samples of the same (large) size are selected from a population, what percentage of all sample proportions will be within \(2.0\) standard deviations of the population proportion?

If all possible samples of the same (large) size are selected from a population, what percentage of all the sample means will be within \(1.5\) standard deviations of the population mean?

Refer to Exercise \(7.100 .\) Seventy percent of adults favor some kind of government control on the prices of medicines. Assume that this percentage is true for the current population of all adults. a, Find the probability that the proportion of adults in a random sample of 400 who favor some kind of government control on the prices of medicincs is i. less than \(.65\) ii. between \(.73\) and \(.76\) b. What is the probability that the proportion of adults in a random sample of 400 who favor some kind of government control is within 06 of the population proportion? c. What is the probability that the sample proportion is greater than the population proportion by \(.05\) or more? Assume that sample includes 400 adults.

Let \(x\) be a continuous random variable that has a normal distribution with \(\mu=75\) and \(\sigma=14 .\) Assuming \(n / N \leq .05\), find the probability that the sample mean, \(\bar{x}\), for a random sample of 20 taken from this population will be a. between \(68.5\) and \(77.3\) b. less than \(72.4\)

The GPAs of all students enrolled at a large university have an approximately normal distribution with a mean of \(3.02\) and a standard deviation of \(.29 .\) Find the probability that the mean GPA of a random sample of 20 students selected from this university is a. \(3.10\) or higher b. \(2.90\) or lower c. \(2.95\) to \(3.11\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

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.