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Lecture 16

STAT 100 Lecture 16: Inference One Sample Proportion

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Statistics and Probability
STAT 100

Week 12 Lecture 16 April 19th, 2017 Inference One Sample Proportion Standardized Sample Proportion Inference about a population proportion p is based on the z statistic that results from standardizing (p-hat): z = (p-hat) — p ÷ √p(1 — p) ÷ n z has approximately the standard normal distribution as long as the sample is not too small and the sample is not a large part of the entire population. Distribution of the Sample Proportion Because the values of the sample proportion varies from sample to sample, it is a random variable. So, we have the same questions for the sample proportion as we had for the sample mean: What is the mean of the sample proportion? p What is the standard deviation of the sample proportion? SE = √p(1 — p) ÷ n What is the sampling distribution of the sample proportion if np (1 — p) ≥ 10 or number of success or failures is 15 or more? (p-hat) ~ N( mean: µ(p-hat) = p, standard error: √p(1 — p) ÷ n) Large-Sample Confidence Interval for a Proportion How do we find the critical value for our confidence interval? statistic +/- (critical value) x (standard deviation of statistic) If the Normal condition is met, we can use a Normal curve. To find a level C confidence interval, we need to catch the central area C under the standard Normal curve. For example, to find a 95% confidence interval, we use a critical value of 2 based on the 68-95-99.7 rule. Using a Standard Normal Table or a calculator, we can get a more accurate critical value. Note, the critical value z* is actually 1.96 for a 95% confidence level. Confidence Interval Draw an SRS of size n from a population with unknown proportion p of successes. An approximate level C confidence interval for p is p +/= z(å/2) √(p-hat) x (1 — p-hat) ÷ n Where z(å/2) is the critical value for the standard Normal density curve with area C between - z(å/2) and z(å/2). Use this interval only when the counts of suc
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