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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
Larson/Farber 4th ed
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
Larson/Farber 4th ed
Sampling Distribution of Sample Means
Sample 5
Sample 2
Population with μ, σ
The sampling distribution consists of the values of the sample means,
Larson/Farber 4th ed
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.
Larson/Farber 4th ed
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.
Larson/Farber 4th ed
Example: Sampling Distribution of Sample Means
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.
Larson/Farber 4th ed
{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}
Example: Sampling Distribution of Sample Means
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
Example: Sampling Distribution of Sample Means
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
Example: Sampling Distribution of Sample Means
e. Find the mean, variance, and standard deviation of the sampling distribution of the sample means.
Solution:
Larson/Farber 4th ed
Example: Sampling Distribution of Sample Means
b. Graph the probability histogram for the population values.
All values have the same probability of being selected (uniform distribution)
Example: Sampling Distribution of Sample Means
f. Graph the probability histogram for the sampling distribution of the sample means.
Larson/Farber 4th ed
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.
Larson/Farber 4th ed
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
Larson/Farber 4th ed
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)
Larson/Farber 4th ed
The Central Limit Theorem
1. Any Population Distribution 2. Normal Population Distribution
Distribution of Sample Means, n ≥ 30
Distribution of Sample Means, (any n)
Larson/Farber 4th ed
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
Larson/Farber 4th ed