Section 5.4 Sampling Distributions and the Central Limit Theorem Larson/Farber 4th ed.

Section 5.4

Sampling Distributions and the Central Limit Theorem

Larson/Farber 4th ed

Section 5.4 Objectives

• Find sampling distributions and verify their properties

• Interpret the Central Limit Theorem• Apply the Central Limit Theorem to find the

probability of a sample mean


Sampling Distributions

Sampling distribution • The probability distribution of a sample statistic. • Formed when samples of size n are repeatedly taken

from a population. • e.g. Sampling distribution of sample means


Sampling Distribution of Sample Means

Sample 5

Sample 2

Population with μ, σ

The sampling distribution consists of the values of the sample means,


Sample 1

2. The standard deviation of the sample means, , is equal to the population standard deviation, σ divided by the square root of the sample size, n.

1. The mean of the sample means, , is equal to the population mean μ.

Properties of Sampling Distributions of Sample Means

• Called the standard error of the mean.


Example: Sampling Distribution of Sample Means

Four people in a carpool paid the following amounts for textbooks this quarter: $120, $140, $180 and $220. Using sample size n = 2 with replacement.

a. Find the mean, variance, and standard deviation of the population.



c. List all the possible samples, with replacement, of size n = 2 and calculate the mean of each sample.

These means form the sampling distribution of sample means.


{120 120, 120 140, 120 180, 120 220, 140 120, 140 140, 140 180, 140 220,180 120, 180 140, 180 180, 180 220,220 120, 220 140, 220 180, 220 220}


Sample Mean

120, 120 120

120, 140 130

120, 180 150

120, 220 170

140, 120 130

140, 140 140

140, 180 160

140, 220 180

Sample Mean

180, 120 150

180, 140 160

180, 180 180

180, 220 200

220, 120 170

220, 140 180

220, 180 200

220, 220 220


d. Construct the probability distribution of the sample means. f Probabili

ty

120 1 120 0.0625 -45 2025 2025

130 2 260 0.125 -35 1225 2450

140 1 140 0.0625 -25 625 625

150 2 300 0.125 -15 225 450

160 2 320 0.125 -5 25 50

170 2 340 0.125 5 25 50

180 3 540 0.1875 15 225 675

200 2 400 0.125 35 1225 2450

220 1 220 0.0625 55 3025 3025

16 2640 11800Larson/Farber 4th ed


e. Find the mean, variance, and standard deviation of the sampling distribution of the sample means.

Solution:



b. Graph the probability histogram for the population values.

All values have the same probability of being selected (uniform distribution)


f. Graph the probability histogram for the sampling distribution of the sample means.


The Central Limit Theorem1. If samples of size n ≥ 30, are drawn from any

population with mean = μ and standard deviation = σ,

x

x

then the sampling distribution of the sample means approximates a normal distribution. The greater the sample size, the better the approximation.


The Central Limit Theorem2. If the population itself is normally distributed,

the sampling distribution of the sample means is normally distribution for any sample size n.

x

x


The Central Limit Theorem

• In either case, the sampling distribution of sample means has a mean equal to the population mean.

• The sampling distribution of sample means has a variance equal to 1/n times the variance of the population and a standard deviation equal to the population standard deviation divided by the square root of n.

Variance

Standard deviation (standard error of the mean)


The Central Limit Theorem

1. Any Population Distribution 2. Normal Population Distribution

Distribution of Sample Means, n ≥ 30

Distribution of Sample Means, (any n)


Example: Interpreting the Central Limit Theorem

The mean age of employees at a large company is 47.2 with a standard deviation of 3.6 years. A random selection of 36 employees is drawn and the mean of each sample is determined. Find the mean and standard error of the mean of the sampling distribution. Then sketch a graph of the sampling distribution of sample means.

Solution: Interpreting the Central Limit Theorem

• The mean of the sampling distribution is equal to the population mean

• The standard error of the mean is equal to the population standard deviation divided by the square root of n.

Solution: Interpreting the Central Limit Theorem

• Since the sample size is greater than 30, the sampling distribution can be approximated by a normal distribution with

Section 5.4 Summary

• Found sampling distributions and verify their properties

• Interpreted the Central Limit Theorem• Applied the Central Limit Theorem to find the

probability of a sample mean


Section 5.4 Sampling Distributions and the Central Limit Theorem Larson/Farber 4th ed.

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Transcript of Section 5.4 Sampling Distributions and the Central Limit Theorem Larson/Farber 4th ed.