/*! 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 21 A certain population has a mean ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 21

A certain population has a mean of 500 and a standard deviation of \(30 .\) Many samples of size 36 are randomly selected and the means calculated. a. What value would you expect to find for the mean of all these sample means? b. What value would you expect to find for the standard deviation of all these sample means? c. What shape would you expect the distribution of all these sample means to have?

Short Answer

Expert verified
a. The expected value of the mean of the sample means is 500. b. The standard deviation of the sample means is expected to be 5. c. The distribution of sample means is expected to have a roughly normal shape.

Step by step solution

01

Find the Mean of Sample Means

The problem asks for the expected mean of all these sample means. According to the Central Limit Theorem, the mean of a sampling distribution is equal to the mean of the population. Therefore, the mean of the sample means is expected to be \(500\) (the given population mean).
02

Find the Standard Deviation of Sample Means

The problem asks for the expected standard deviation of the sample means. The standard deviation of a sampling distribution, also known as the standard error, is given by \(\sigma / \sqrt{n}\), where \(\sigma\) is the standard deviation of the population and \(n\) is the size of each sample. Using the given values from the problem, this would be calculated as follows: \(30 / \sqrt{36} = 30 / 6 = 5\). Therefore, the standard deviation of the sample means is expected to be \(5\).
03

Determine the Shape of the Distribution

The Central Limit Theorem tells us that as the sample size gets larger, the distribution of the sample means approaches a normal distribution, regardless of the shape of the population distribution. Considering our sample size of 36 is large enough, we would expect the distribution of sample means to be approximately normal. This is true no matter what the shape of the original population distribution.

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
A sampling distribution is a statistical concept that focuses on understanding how sample statistics represent a population. When you take multiple random samples from a population, each sample has its own mean, and these means form a distribution. This is called a sampling distribution of the sample means.
  • The samples are taken randomly, ensuring that each has the same chance of being selected.
  • The sampling distribution provides insight into the variation that arises when these sample statistics are analyzed.
  • It helps estimate population parameters, like the population mean and standard deviation, by providing a distribution mean and standard deviation.
Understanding sampling distribution helps in making inferences about a whole population based on the statistics derived from samples.
Mean of Sample Means
The mean of sample means, often referred to as the expected value of sample means, is a critical concept in statistics. According to the Central Limit Theorem, the average of these sample means will always equal the mean of the population if the sample size is large enough.
  • This implies that the mean of all possible sample means gives you the population mean.
  • For instance, if the population mean is 500, as in our problem, the mean of the sample means, no matter how many samples you take, will also be 500.
  • This allows for accurate predictions and inferences about the population characteristics.
This result is independent of the distribution of the population itself, showcasing the strength of the Central Limit Theorem.
Standard Deviation of Sample Means
The standard deviation of the sample means, known as the standard error, measures the dispersion in the sample means around the population mean. It is calculated using the formula \(\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.
  • This value helps determine how much sample means tend to deviate from the actual population mean.
  • In our exercise, with a population standard deviation of 30 and sample size of 36, the standard deviation of the sample means is \(\frac{30}{6} = 5\).
  • The smaller this number, the closer the sample means are to the population mean, indicating more reliable samples.
This measure is crucial for assessing the precision of sampling processes in estimating population parameters.
Normal Distribution
Normal distribution, often represented as a bell curve, is a fundamental shape in statistics due to its unique properties. The Central Limit Theorem states that the distribution of sample means will approximate a normal distribution as the sample size increases, regardless of the population's initial distribution shape.
  • This approximation is valid for a sample size of 30 or more, making 36 a suitable size for expecting a normal distribution.
  • The normal distribution of sample means allows for easier statistical analysis and predictions.
  • Even if the population is not normally distributed, the sample means will be, making statistical inference easier and more reliable.
This can simplify hypothesis testing and confidence interval calculations, making normal distribution a key aspect of statistical assessments.

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

TIMSS 2007 (Trends in International Mathematics and Science Study) focused on the mathematics and science achievement of eighth-grade students throughout the world. A total of 8 countries (including the United States) participated in the study. The mean math exam score for U.S. students was 509 with a standard deviation of \(88 .\) Assuming the scores are normally distributed, find the following for a sample of 150 students. a. Find the probability that the mean TIMSS score for a randomly selected group of eighth-grade students would be between 495 and 515. b. Find the probability that the mean TIMSS score for a randomly selected group of eighth-grade students would be less than 520. c. Do you think the assumption of normality is reasonable? Explain.

The heights of the kindergarten children mentioned in Example 7.6 (p. 328 ) are approximately normally distributed with \(\mu=39\) and \(\sigma=2\). a. If an individual kindergarten child is selected at random, what is the probability that he or she has a height between 38 and 40 inches? b. \(\quad\) A classroom of 30 of these children is used as a sample. What is the probability that the class mean \(\bar{x}\) is between 38 and 40 inches? c. If an individual kindergarten child is selected at random, what is the probability that he or she is taller than 40 inches? d. A classroom of 30 of these kindergarten children is used as a sample. What is the probability that the class mean \(\bar{x}\) is greater than 40 inches?

Consider the approximately normal population of heights of male college students with mean \(\mu=69\) inches and standard deviation \(\sigma=4\) inches. A random sample of 16 heights is obtained. a. Describe the distribution of \(x,\) height of male college students. b. Find the proportion of male college students whose height is greater than 70 inches. c. Describe the distribution of \(\bar{x},\) the mean of samples of size 16. d. Find the mean and standard error of the \(\bar{x}\) distribution. e. \(\quad\) Find \(P(\bar{x}>70)\). f. \(\quad\) Find \(P(\bar{x}<67)\).

According to the June 2004 Readers' Digest article "Only in America," the average amount that a 17 -year old spends on his or her high school prom is \(638\) Assume that the amounts spent are normally distributed with a standard deviation of \(175\) a. Find the probability that the mean cost to attend a high school prom for 36 randomly selected 17-year-olds is between \(550\) and \(700\). b. Find the probability that the mean cost to attend a high school prom for 36 randomly selected 17 -yearolds is greater than \(\$ 750\). c. Do you think the assumption of normality is reasonable? Explain.

Salaries for various positions can vary significantly, depending on whether or not the company is in the public or private sector. The U.S. Department of Labor posted the 2007 average salary for human resource managers employed by the federal government as \( 76,503 .\) Assume that annual salaries for this type of job are normally distributed and have a standard deviation of \(8850\) a. What is the probability that a randomly selected human resource manager received over \(100,000\) in \(2007 ?\) b. \(\quad\) A sample of 20 human resource managers is taken and annual salaries are reported. What is the probability that the sample mean annual salary falls between \(70,000\) and \(80,000 ?\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

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.