1. Populations and Samples Central Limit Theorem

2. Lecture Objectives You should be able to: 1.Define the Central Limit Theorem 2.Explain in your own words the relationship between a population distribution and the distribution of the sample means.

3. The Population X = The incomes of all working residents of a town The population size is 10,000. Refer to Central Limit.xls for the population data.Central Limit.xls

4. Population Distribution Mean $50,185.85 Stdevp $28,772.27 Note that the distribution is uniform, not normal

5. Samples (n=36) Sample 1 50 samples of size 36 each are taken from this population. The distributions of the first 3 samples are shown. How do they compare to the population? Mean $54,628.06 Stdev $26,122.75

6. Sample 2 Mean $41,987.92 Stdev $27,950.33

7. Sample 3 Mean $52,875.11 Stdev $26,939.75

8. Sample Means Sample NumberMeanNumberMeanNumberMeanNumberMeanNumberMean 154628.061155104.692145861.723156073.614155463.14 241987.921249068.922250664.943252910.084260488.08 352875.111351828.392347606.473342266.004350382.19 450518.611456782.642452480.003445048.754454254.17 552685.441547663.692553563.223555515.644548620.89 651243.831650070.112646180.893652098.584643133.47 740256.191751850.222746961.083749449.814748488.06 848968.671855989.332856496.503839071.424848064.61 949881.921946046.722944940.893946978.504948492.58 1050413.532050986.033056167.114052044.035045618.28 The means of 50 such samples of size 36 each are shown below.

9. Distribution of Sample Means Mean50084.70 Stdev4607.82

10. Population Mean = =50,185.85 Mean of Sample Means = = 50,084.70 Population Standard Deviation = =28,772.27 Standard Deviation of Sample Means = = 4,607.82 (also called Standard Error, or SE) (Pop. Standard Deviation) / SE = 6.24 Sample size (n) = 36 Square root of sample size √ n = 6 Population and Sampling Means

11. Central Limit Theorem Regardless of the population distribution, the distribution of the sample means is approximately normal for sufficiently large sample sizes (n>=30), with and

12. Questions 1.How will the distribution of sample means change if the sample size goes up to n=100? the sample size goes down to n=2? 2.Is the distribution of a single sample the same as the distribution of the sample means? 3.If a population mean = 100, and pop. standard deviation = 24, and we take all possible samples of size 64, the mean of the sampling distribution (sample means) is _______ and the standard deviation of the sampling distribution is _______.

13. Applying the results If the sample means are normally distributed, what proportion of them are within ± 1 Standard Error? what proportion of them are within ± 2 Standard Errors? If you take just one sample from a population, how likely is it that its mean will be within 2 SEs of the population mean? How likely is it that the population mean is within 2 SEs of your sample mean?

14. The population mean is within 2 SEs of the sample mean, 95% of the time. Thus, is in the range defined by: 2*SE, about 95% of the time. (2 *SE) is also called the Margin of Error (MOE). 95% is called the confidence level. Confidence Intervals