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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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 }

8. Example: Sampling Distribution of Sample Means SampleMean 120, 120120 120, 140130 120, 180150 120, 220170 140, 120130 140, 140140 140, 180160 140, 220180 SampleMean 180, 120150 180, 140160 180, 180180 180, 220200 220, 120170 220, 140180 220, 180200 220, 220220

9. Example: Sampling Distribution of Sample Means d. Construct the probability distribution of the sample means. fProbabili ty 1201 0.0625-452025 1302 2600.125-3512252450 1401 0.0625-25625 1502 3000.125-15225450 1602 3200.125-52550 1702 3400.12552550 1803 5400.187515225675 2002 4000.1253512252450 2201 0.0625553025 16264011800 Larson/Farber 4th ed

10. 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

11. 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)

12. Example: Sampling Distribution of Sample Means f. Graph the probability histogram for the sampling distribution of the sample means. Larson/Farber 4th ed

13. The Central Limit Theorem 1. If samples of size n  30, are drawn from any population with mean =  and standard deviation = , 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

14. The Central Limit Theorem 2. 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

15. 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

16. The Central Limit Theorem 1.Any Population Distribution2.Normal Population Distribution Distribution of Sample Means, n ≥ 30 Distribution of Sample Means, (any n) Larson/Farber 4th ed

17. 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.

18. 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.

19. 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

20. 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