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Chapter 8: Problem 34

Load the sampling distribution applet on your computer. Set the applet so that the population is bell shaped. Take note of the mean and standard deviation. (a) Obtain 1000 random samples of size \(n=5 .\) Describe the distribution of the sample mean based on the results of the applet. According to statistical theory, what is the distribution of the sample mean? (b) Obtain 1000 random samples of size \(n=10 .\) Describe the distribution of the sample mean based on the results of the applet. According to statistical theory, what is the distribution of the sample mean? (c) Obtain 1000 random samples of size \(n=30 .\) Describe the distribution of the sample mean based on the results of the applet. According to statistical theory, what is the distribution of the sample mean? (d) Compare the results of parts (a)-(c). How are they the same? How are they different?

Short Answer

Expert verified
As sample size increases from 5 to 30, the sample mean distribution becomes closer to normal and has a smaller standard deviation.

Step by step solution

01

Setting Up the Applet

Load the sampling distribution applet on your computer and set the population to be bell-shaped. Note the population mean (渭) and standard deviation (蟽).
02

Obtain 1000 Random Samples of Size n=5

Generate 1000 random samples where each sample has a size of n=5. Record the sample means. Describe the distribution shape, center, and spread of these sample means. According to the Central Limit Theorem, the distribution of the sample mean will approximate a normal distribution with mean 渭 and standard deviation \[ \frac{\text{蟽}}{\text{鈭歯}} \] where n is the sample size (n=5 in this case).
03

Obtain 1000 Random Samples of Size n=10

Generate 1000 random samples where each sample has a size of n=10. Record the sample means. Describe the distribution shape, center, and spread of these sample means. According to the Central Limit Theorem, the distribution of the sample mean will approximate a normal distribution with mean 渭 and standard deviation \[ \frac{\text{蟽}}{\text{鈭歯}} \] where n is the sample size (n=10 in this case).
04

Obtain 1000 Random Samples of Size n=30

Generate 1000 random samples where each sample has a size of n=30. Record the sample means. Describe the distribution shape, center, and spread of these sample means. According to the Central Limit Theorem, the distribution of the sample mean will approximate a normal distribution with mean 渭 and standard deviation \[ \frac{\text{蟽}}{\text{鈭歯}} \] where n is the sample size (n=30 in this case).
05

Compare Results

Compare the distributions of the sample means obtained in parts (a), (b), and (c). Note how the sample size affects the shape, center, and spread of the distributions. Larger sample sizes should result in distributions closer to normal with smaller standard deviations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sampling Distribution
When you repeatedly take samples from a population and compute a statistic, like the mean, the distribution of these statistics is called the sampling distribution. A key idea here is that it's not just one sample, but many samples. For example, if you take 1000 samples of size 5 from a population, you'll get 1000 sample means. The shape and other characteristics of these sample means form the sampling distribution.
In the exercise, you used an applet to see the sampling distribution for different sample sizes. This helps you understand how sample size affects the distribution of the sample means.
Sample Mean
The sample mean is simply the average of the observations in your sample. Suppose you take a sample of size 5 from a population. You calculate the sample mean by adding up all the values in your sample and then dividing by 5. If you take many samples (like 1000), you'll get 1000 sample means.
According to statistical theory, specifically the Central Limit Theorem (CLT), the distribution of sample means will form a normal distribution if the sample size is large enough. Even if the original population is not normally distributed, the sample means will tend to be normally distributed as the sample size increases.
Normal Distribution
A normal distribution is a bell-shaped distribution that is symmetrical about the mean. It's one of the most common distributions in statistics. Many statistical tests are based on the assumption of normality.
The Central Limit Theorem (CLT) tells us that the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the original population distribution, as long as the sample size is large enough. In the exercise, you saw this property in action: as the sample size increased (from 5 to 10 to 30), the distribution of the sample means became more like a normal distribution.
Standard Deviation
Standard deviation measures the spread or variability of a distribution. For a population, it tells you how much the individual observations differ from the mean. For the sampling distribution of the sample mean, the standard deviation tells you how much the sample means differ from the population mean.
According to the Central Limit Theorem, the standard deviation of the sampling distribution of the sample mean is given by \[ \frac{\text{蟽}}{\text{鈭歯}} \] where 蟽 is the standard deviation of the population and n is the sample size. This means that as the sample size increases, the standard deviation of the sample mean decreases, making the sample means cluster more tightly around the population mean.
In the exercise, when you increased the sample size from 5 to 10 to 30, you would have noticed that the spread of the sample means became smaller and closer to the population mean.

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Most popular questions from this chapter

Suppose a simple random sample of size \(n=49\) is obtained from a population with \(\mu=80\) and \(\sigma=14\) (a) Describe the sampling distribution of \(\bar{x}\). (b) What is \(P(\bar{x}>83) ?\) (c) What is \(P(\bar{x} \leq 75.8) ?\) (d) What is \(P(78.3<\bar{x}<85.1) ?\)

The Food and Drug Administration sets Food Defect Action Levels (FDALs) for some of the various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL for insect filth in peanut butter is 3 insect fragments (larvae, eggs, body parts, and so on) per 10 grams. A random sample of 50 ten-gram portions of peanut butter is obtained and results in a sample mean of \(\bar{x}=3.6\) insect fragments per ten-gram portion. (a) Why is the sampling distribution of \(\bar{x}\) approximately normal? (b) What is the mean and standard deviation of the sampling distribution of \(\bar{x}\) assuming \(\mu=3\) and \(\sigma=\sqrt{3}\) (c) Suppose a simple random sample of \(n=50\) ten-gram samples of peanut butter results in a sample mean of 3.6 insect fragments. What is the probability a simple random sample of 50 ten-gram portions results in a mean of at least 3.6 insect fragments? Is this result unusual? What might we conclude?

Burger King's Drive-Through Suppose cars arrive at Burger King's drive-through at the rate of 20 cars every Hour between 12: 00 noon and 1: 00 P.M. A random sample \- of 40 one-hour time periods between 12: 00 noon and 1: 00 P.M. is selected and has 22.1 as the mean number of cars erriving. (a) Why is the sampling distribution of \(\bar{x}\) approximately normal? (b) What is the mean and standard deviation of the sampling distribution of \(\bar{x}\) assuming \(\mu=20\) and \(\boldsymbol{\sigma}=\sqrt{20}\) (c) What is the probability that a simple random sample of 40 one-hour time periods results in a mean of at least 22.1 cars? Is this result unusual? What might we conclude?

The following data represent the ages of the winners of the Academy Award for Best Actor for the years \(1999-2004\) $$\begin{array}{lr} 2004: \text { Jamie Foxx } & 37 \\ \hline 2003 \text { - Sean Penn } & 43 \\ \hline 2002 \text { . Adrien Brody } & 29 \\ \hline 2001: \text { Denzel Washington } & 47 \\ \hline 2000 \text { . Russell Crowe } & 36 \\ \hline 1999 \text { . Kevin Spacey } & 40 \end{array}$$ (a) Compute the population mean, \(\mu\) (b) List all possible samples with size \(n=2\). There should be \(_{6} C_{2}=15\) samples. (c) Construct a sampling distribution for the mean by listing the sample means and their corresponding probabilities. (d) Compute the mean of the sampling distribution. (e) Compute the probability that the sample mean is within 3 years of the population mean age. (f) Repeat parts \((b)-(e)\) using samples of size \(n=3\) Comment on the effect of increasing the sample size.

Suppose a simple random sample of size \(n=75\) is obtained from a population whose size is \(N=10,000\) and whose population proportion with a specified characteristic is \(p=0.8\) (a) Describe the sampling distribution of \(\hat{p}\) (b) What is the probability of obtaining \(x=63\) or more individuals with the characteristic? That is, what is \(P(\hat{p} \geq 0.84) ?\) (c) What is the probability of obtaining \(x=51\) or fewer individuals with the characteristic? That is, what is \(P(\hat{p} \leq 0.68) ?\)
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