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Chapters 18 and 19

# Chapters 18 and 19 - Chapter 18 Sampling Distribution...

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Chapter 18 Sampling Distribution Models The retirement fund in a company is invested in five corporate stocks with the following returns: Stock ........................ Return A ................................. 7% B ................................ 12% C ................................. -3% D ................................ 21% E .................................. 3%

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Chapter 18 Sampling Distribution Models The population mean = 8% The population standard deviation =8.15%. Take a random sample of 3 stocks. Assuming that the order is not important and sampling is done without replacement, applying combination equation (n=5, and x=3) there are ten possibilities:
Chapter 18 Sampling Distribution Models Sample Stocks ............... Returns ............. Mean 1) A, B, C ...................... 7%..12%..-3% ...... 5.33% 2) A, B, D ...................... 7%..12%..21% .... 13.33% 3) A, B, E ...................... 7%..12%..3% ........ 7.33% 4) A, C, D ...................... 7%..-3%..21% ...... 8.33% 5) A, C, E ...................... 7%..-3%..3% ........ 2.33% 6) A, D, E ...................... 7%..21%..3% ...... 10.33% 7) B, C, D ..................... 12%..-3%..21% ..... 10.00% 8) B, C, E ..................... 12%..-3%..3% ........ 4.00% 9) B, D, E ..................... 12%..21%..3% ...... 12.00% 0) C, D, E ..................... -3%..21%..3% ........ 7.00%

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Chapter 18 Sampling Distribution Models Two (or more) samples from the same population will likely have different sample values (mean values ranges from 2.33% to 13.33%), and therefore possibly lead to different decisions. The sample means also are different from the population mean, i.e., 8. For example, if sample 4 is selected, the sampling error (the difference between a sample statistic and its corresponding population parameter) is fairly small (8.33 - 8.0 = 0.33) but if the selected sample is sample 2, the error is quite large (13.33 - 8.0 = 5.33). Because the decision maker cannot know how large the sampling error will be before selecting the sample, he/she should know how the possible sample means are distributed.
Chapter 18 Sampling Distribution Models Definition: The distribution of all possible sample means and their related probability is called the sampling distribution of the means. Properties of the Sampling Distribution of Means: If a population is normally distributed, then: 1. The mean of the sampling distribution of means equals the population mean. 2. The standard deviation of the sampling distribution of means (or standard error of the mean) is smaller than the population standard deviation

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Chapter 18 Sampling Distribution Models The Central Limit Theorem: As the sample size n, increases, the mean of n independent values has a sampling distribution that tends towards a Normal model. Its mean equals the mean of the population mean Its standard deviation equals the standard deviation of the population divided by square root of the sample size
Chapter 18 Sampling Distribution Models

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Chapter 18 Sampling Distribution Models means that the mean of all means is the population mean. means that standard deviation of all the means is reduced by a factor of square root of n.
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Chapters 18 and 19 - Chapter 18 Sampling Distribution...

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