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.

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.

Population Distribution

Histogram of Population Data

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Mean $50,185.85

Stdevp $28,772.27

Note that the distribution is uniform, not normal

Samples (n=36) Sample 1

Histogram Sample 1

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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

Histogram Sample 2

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Sample 2

Mean $41,987.92

Stdev $27,950.33

Sample 3

Histogram Sample 3

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Mean $52,875.11

Stdev $26,939.75

Sample Means

Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample

Number Mean Number Mean Number Mean Number Mean Number Mean

1 54628.06 11 55104.69 21 45861.72 31 56073.61 41 55463.14

2 41987.92 12 49068.92 22 50664.94 32 52910.08 42 60488.08

3 52875.11 13 51828.39 23 47606.47 33 42266.00 43 50382.19

4 50518.61 14 56782.64 24 52480.00 34 45048.75 44 54254.17

5 52685.44 15 47663.69 25 53563.22 35 55515.64 45 48620.89

6 51243.83 16 50070.11 26 46180.89 36 52098.58 46 43133.47

7 40256.19 17 51850.22 27 46961.08 37 49449.81 47 48488.06

8 48968.67 18 55989.33 28 56496.50 38 39071.42 48 48064.61

9 49881.92 19 46046.72 29 44940.89 39 46978.50 49 48492.58

10 50413.53 20 50986.03 30 56167.11 40 52044.03 50 45618.28

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

Distribution of Sample Means

Distribution of the Sample Means50 samples of size 36 each

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Sample Means

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Mean 50084.70

Stdev 4607.82

Population Mean = = 50,185.85Mean of Sample Means = = 50,084.70

Population Standard Deviation = = 28,772.27Standard Deviation of Sample Means = = 4,607.82(also called Standard Error, or SE)

(Pop. Standard Deviation) / SE = 6.24

Sample size (n) = 36Square root of sample size √n = 6

Population and Sampling Means

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Central Limit Theorem

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Regardless of the population distribution, the distribution of the sample means is approximately normal for sufficiently large sample sizes (n>=30), with

and

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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

x