6 - STAT E-50 Introduction to Statistics Sampling...

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STAT E-50 - Introduction to Statistics Sampling Distribution Models; Confidence Intervals for Proportions Notation: p Proportion, or probability of “success” in the model q Probability of “failure” in the model (p + q = 1, so q = 1 - p) ˆ p Observed proportion in the data ˆ q Observed proportion of “failure” in the data ( ) ˆˆ q1p = Page 1

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Page 2 The Sampling Distribution of a Sample Statistic The sampling distribution model shows the behavior of the statistic over all possible samples of a population. The samples must all be the same size, and the sample statistic can be any descriptive sample statistic.
The Sampling Distribution Model for a Proportion Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of ˆ p is modeled by a Normal model with ± mean ˆ μ (p) p = ± standard deviation pq ˆ SD(p) n = Page 3

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Page 4 Assumptions: The sampled values must be independent of each other The sample size, n, must be large enough
Conditions you can check: 1. 10% Condition If sampling is not with replacement, the sample size, n, must be no larger than 10% of the population. 2. Success/Failure Condition The sample size must be large enough so that np 10 and nq 10 Page 6

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1. The development of viral hepatitis subsequent to a blood transfusion can cause serious complications. The article “Hepatitis in Patients with Acute Non-lymphatic Leukemia” ( Amer. J. of Med. (1983): 413-421) reported that in spite of careful screening for those having an hepatitis antigen, viral hepatitis occurs in 7% of blood recipients. Here are the results of several simulations of sampling with 500 samples each, of different sample sizes, using p = .07: Page 7
1. The development of viral hepatitis subsequent to a blood transfusion can cause serious complications. The article “Hepatitis in Patients with Acute Non-lymphatic Leukemia” ( Amer. J. of Med. (1983): 413-421) reported that in spite of careful screening for those having an hepatitis antigen, viral hepatitis occurs in 7% of blood recipients. Page 8

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Page 9 2. Assume that the proportion of all blood recipients stricken with viral hepatitis is .07. Suppose that a new treatment is believed to reduce the incidence of viral hepatitis. This treatment is given to 200 blood recipients, only 6 of whom contract hepatitis. Does this result indicate that the true (long-run) proportion of patients who contract hepatitis after the experimental treatment is less than .07, or could this result be plausibly attributed to sample variability? What do we want to know? If the treatment is ineffective, how likely is it that only 6 (or fewer) of the 200 patients would have contracted viral hepatitis? Check the conditions:
Page 10 2. Assume that the proportion of all blood recipients stricken with viral hepatitis is .07.

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6 - STAT E-50 Introduction to Statistics Sampling...

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