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Unformatted text preview: Chapter 8 PROPORTIONS Many statistical studies produce counts rather than measurements. Example: Did you vote in the last election? The response would be either a Yes or a No. If I did a survey, I would accumulate the count of Yes responses and describe this count as a proportion of the total. Proportion that voted = = 338/ 1500= .225 Example: What academic year are you in at Purdue. The response would be either Freshman, Sophomore, Junior, or Senior. Again, I could accumulate the count of each and describe each as a proportion of the total. Proportion of freshmen = .2045 Proportion of sophomores = .2688 Proportion of juniors = .3215 Proportion of seniors = .2052 Population and Sample proportions: In statistical sampling we often want to estimate the proportion, p, of successes in a population. Success is when the categorical variable takes on one particular value of interest. p = count of successes in population size of population = X / N = a fixed but unknown value between 0 and 1.0 When we take a sample of our population ; our estimator is the sample proportion of successes: p = count of successes in sample size of sample Page 1 = X / n = a known value but another sample would give a different value. Example: 1. You flip a coin 20 times and record whether a head (success) or a tail is tossed. In this sample, a head is recorded 11 times. What is the sample proportion of heads? = 11/20 = .55 Inference for a Single Proportion: So far we have only looked at making a statistical inference on population means. Now we will look at making a statistical inference on a population proportion. Examples: A newspaper runs a poll to determine the popularity of a congressman in the district. The poll results in a value = .52. In this example we are interested in estimating the unknown proportion p in the population of all voters in the district. The point estimate of that population parameter p is the sample proportion p , a statistic. Sampling Distribution of a Sample Proportion: If you select a SRS of size n from a large population, and determine the sample proportion in favor of a certain issue, p = X/ n, you will find that: The sample proportion is a statistic and fluctuates, The sampling distribution of p is approximately normal provided the sample size is big enough to produce at least 15 success and 15 failures, and does not exceed 1% of the population....
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- Spring '08