Central Limit Theorem Section 5-5 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

As the sample size increases the distribution of sample means will approach a normal distribution.

(Last 4 digits from 50 students) Distribution of 200 digits From Social Security Numbers (Last 4 digits from 50 students) 20 Frequency 10 0 1 2 3 4 5 6 7 8 9 Distribution of 200 digits Figure 5-19

Table 5-1 SSN digits x 1 5 9 4 7 8 3 1 2 7 6 3 8 2 1 5 4 4 6 8 5 2 9 4.75 4.25 8.25 3.25 5.00 3.50 5.25 2 6 5 7 8 3 7 4 5 1 6 2 7 3 8 6 2 6 1 9 5 7 8 4 4.00 5.25 4.25 4.50 4.75 3.75 6.00

Distribution of 50 Sample Means for Students 15 Frequency 10 5 0 1 2 3 4 5 6 7 8 9 Distribution of 50 Sample Means Figure 5-20

Normal, Uniform, and Skewed Distributions Figure 5-24 Normal Skewed Uniform Original population Sample means ( n = 5 ) Sample means ( n = 10 ) Sample means ( n = 30 )

Normal, Uniform, and Skewed Distributions Figure 5-24 Uniform Normal Original population Sample means ( n = 5 ) Sample means ( n = 10 ) Sample means ( n = 30 )

Normal, Uniform, and Skewed Distributions Figure 5-24 Skewed Normal Uniform Original population Sample means ( n = 5 ) Sample means ( n = 10 ) Sample means ( n = 30 )

Central Limit Theorem Given: 1. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation s. 2. Samples of size n are randomly selected from this population.

Central Limit Theorem Given: Conclusions: 1. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation s. 2. Samples of size n are randomly selected from this population. Conclusions:

Central Limit Theorem Given: Conclusions: 1. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation s. 2. Samples of size n are randomly selected from this population. Conclusions: 1. The distribution of sample means x will, as the sample size increases, approach a normal distribution.

Central Limit Theorem Given: Conclusions: s — n  1. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation s. 2. Samples of size n are randomly selected from this population. Conclusions: 1. The distribution of sample means x will, as the sample size increases, approach a normal distribution. 2. The mean of the sample means will be the population mean µ. 3. The standard deviation of the sample means will be s — n 

Practical Rules Commonly Used : 1. For samples of size n larger than 30, the distribution of the sample means can be approximated reasonably well by a normal distribution. The approximation gets better as the sample size n becomes larger. 2. If the original population is itself normally distributed, then the sample means will be normally distributed for any sample size n.

Notation

the standard deviation of sample mean Notation the mean of the sample means µx = µ the standard deviation of sample mean sx = (often called standard error of the mean) s n

the standard deviation of sample means Notation the mean of the sample means µx = µ the standard deviation of sample means sx = (often called standard error of the mean) s n

the standard deviation of sample mean Notation the mean of the sample means µx = µ the standard deviation of sample mean sx = (often called standard error of the mean) s n

Sampling Without Replacement If n > 0.05 N

Sampling Without Replacement If n > 0.05 N N – n s sx = n N – 1

Sampling Without Replacement If n > 0.05 N N – n s sx = n N – 1 finite population correction factor