Populations and Samples Central Limit Theorem. Lecture Objectives You should be able to: 1.Define the Central Limit Theorem 2.Explain in your own words.

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Presentation transcript:

Populations and Samples Central Limit Theorem

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.

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

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

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, Stdev $26,122.75

Sample 2 Mean $41, Stdev $27,950.33

Sample 3 Mean $52, Stdev $26,939.75

Sample Means Sample NumberMeanNumberMeanNumberMeanNumberMeanNumberMean The means of 50 such samples of size 36 each are shown below.

Distribution of Sample Means Mean Stdev

Population Mean = =50, Mean of Sample Means = = 50, Population Standard Deviation = =28, Standard Deviation of Sample Means = = 4, (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

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

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 _______.

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?

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