Chapter 8a Sampling Distributions

Chapter 8a Sampling Distributions - Click to edit Master...

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Unformatted text preview: Click to edit Master subtitle style Professor Thomas R. Sexton Professor Thomas R. Sexton 11 Sampling Distributions Professor Thomas R. Sexton College of Business Stony Brook University Professor Thomas R. Sexton Professor Thomas R. Sexton 22 Estimation Problem What proportion of Stony Brook students smoke? Very large population ( N 23,000). Sample size = n = 100 students X = 20 students smoke Professor Thomas R. Sexton Professor Thomas R. Sexton 33 How Precise? How precise is our estimate of 20%? It is subject to sampling error. If we choose another sample of size 100, we are likely to get a different estimate. How much can these estimates vary? We need to know before we make decisions based on this estimate. Professor Thomas R. Sexton Professor Thomas R. Sexton 44 Using the Binomial Suppose that 25% of all Stony Brook students smoke. What is the probability that we observe exactly 20 smokers in a sample of 100 students? Use the binomial distribution to find the P( X = 20) = P( p = 0.2) = 0.0493. What about all the other possible values of p ? Professor Thomas R. Sexton Professor Thomas R. Sexton 55 The Distribution of p with 100 Students 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p = X/n Probability Professor Thomas R. Sexton Professor Thomas R. Sexton 66 Some Observations The distribution of p looks very much like a normal distribution. l It appears symmetric. l It is centered at 0.25, the true proportion of smokers. The probability of observing p = 0.2 is 0.0493. The probability of observing p 0.2 is 0.1488. Professor Thomas R. Sexton Professor Thomas R. Sexton 77 Suppose We Sample 1000 Students To obtain p = 0.2, we must have 200 smokers in our sample. Now use the binomial distribution with n = 1000 and X = 200. We find that P( X = 200) = P( p = 0.2) = 0.0000287, only 287 times in 10 million. We find that P( p 0.2) = 0.000109, roughly one time in 10,000. Professor Thomas R. Sexton Professor Thomas R. Sexton 88 The Distribution of p with 1000 Students 0.005 0.01 0.015 0.02 0.025 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p = X/n Probability Professor Thomas R. Sexton Professor Thomas R. Sexton 99 Both Distributions 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p = X/n Probability 100 Students 1000 Students Professor Thomas R. Sexton Professor Thomas R. Sexton 1010 Convert to Probability Densities (Divide by 0.01 or 0.001) 5 10 15 20 25 30 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 p = X/n Probability Density 100 Students 1000 Students Professor Thomas R....
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Chapter 8a Sampling Distributions - Click to edit Master...

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