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math concepts notes lecture 25

math concepts notes lecture 25 - Lectur e 25 Reading s...

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Lectur e Reading s Objectives Assignment 25 8.1-8.5 Sampling distributions of sample proportion and sample mean Calculations with sampling distributions Stats: P526 1-15 odd P551 17-25 odd DEFINITION: The sampling distribution of the sample proportion is the distribution of values of the sample proportion in all possible random samples of the same size n taken from the same population. Example Proportion of Women We assume that 50% of the population are women, p = 0.50. Take a simple random sample of size n =4 people from this population and observe the proportion of women in the sample. Then this sampling process will be repeated many times to examine the possible values for the sample proportion and see how these possible values vary. ( seed 91, 1= woman, and 0= man) NUMBER OF WOMEN SAMPLE PROPORTION TALLY FREQUENCY PROPORTION OF ALL TRIALS 0 1 2 3 4 0.00 50 0.25 0.75 0.50 1.00 4 16 10 16 4 4/50=.08 4/50=.08 16/50=.32 16/50=.32 10/50=.20 50/50=1.00 Chapter 8 --- 1
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Properties of the Distribution of the Sample Proportion Chapter 8 --- 2 If simple random samples of size n are taken from a population where the proportion of "successes" is p , then the sampling distribution of p ˆ has the following properties: 1. ( ) E p p = ) The average of all possible p ˆ values is equal to the parameter p . In other words, p ˆ is an unbiased estimator of p . 2. (1 ) ( ) p p p n σ - = ) The standard deviation for p ˆ decreases as the sample size n increases. 3. If n is “sufficiently” large , the distribution of p ˆ eventually looks like a normal distribution with mean and standard deviation as given in 1 and 2 above.
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