316 CHAPTER 7 Estimating Parameters and Determining Sample Sizes proportion of undergraduate students who take online courses is actually p = 0.50. See Figure 7-3, which shows that 19 out of 20 (or 95%) different confidence intervals contain the assumed value of p = 0.50. Figure 7-3 is trying to tell this story: With a 95% confidence level, we expect about 19 out of 20 confidence intervals (or 95%) to contain the true value of p. This confidence interval does not contain p5 0.50. 0.45 p5 0.50 0.55 FIGURE 7-3 Confidence Intervals from 20 Different Samples Confidence Interval for Estimating a Population Proportion p Objective Construct a Wald confidence interval used to estimate a population proportion p. Notation p = population proportion pn = sample proportion n = number of sample values E = margin of error za>2 = critical value: the z score separating an area of a>2 in the right tail of the standard normal distribution Requirements 1. The sample is a simple random sample. 2. The conditions for the binomial distribution are satisfied: There is a fixed number of trials, the trials are independent, there are two categories of outcomes, and the probabilities remain constant for each trial (as in Section 5-2). 3. There are at least 5 successes and at least 5 failures. (This requirement is a way to verify that np Ú 5 and nq Ú 5, so the normal distribution serves as a suitable approximation to the binomial distribution, which is the distribution of proportions.) (If this requirement is not satisfied, one alternative is to use the method of bootstrap resampling described in Section 7-4.) Confidence Interval Estimate of p and the Margin of Error pn - E 6 p 6 pn + E where the margin of error is E = z a>2Bpnqn n The confidence interval is often expressed in the following two equivalent formats: pn { E or 1p n - E, pn + E2 Round-Off Rule for Confidence Interval Estimates of p Round the confidence interval limits for p to three significant digits. KEY ELEMENTS p S co 9 Go Figure $1,000,000: Estimated difference in lifetime earnings for someone with a college degree and someone without a college degree.
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