318 CHAPTER 7 Estimating Parameters and Determining Sample Sizes Manual Calculation Here is how to find the confidence interval with manual calculations: a. The margin of error is found by using Formula 7-1 with za>2 = 1.96, pn = 0.53, qn = 0.47, and n = 950. E = za>2 Bpnqn n = 1.96 B10.532 10.472 950 = 0.0317381 b. Constructing the confidence interval is really easy now that we know that pn = 0.53 and E = 0.0317381. Simply substitute those values to obtain this result: pn - E 6 p 6 pn + E 0.53 - 0.0317381 6 p 6 0.53 + 0.0317381 0.498 6 p 6 0.562 1rounded to three significant digits2 This same result could be expressed in the format of 0.53 { 0.032 or 10.498, 0.5622. If we want the 95% confidence interval for the true population percentage, we could express the result as 49.8,6 p 6 56.2,. Note that these expressions resulting from manual calculations differ slightly from the expression obtained using technology. The reason for the discrepancy is that the manual calculations are based on the critical value of z 0.025 = 1.96, but technology uses the more accurate critical value of z 0.025 = 1.959963986. c. Based on the confidence interval obtained in part (b), we cannot safely conclude that more than 50% of undergraduates take online courses. Because the confidence interval ranges from 0.499 to 0.562, it is possible that the population percentage is below 50%. d. Here is one statement that summarizes the results: 53% of undergraduates take online courses. That percentage is based on a Sallie Mae survey of 950 randomly selected undergraduates. In theory, in 95% of such polls, the percentage should differ by no more than 3.2 percentage points in either direction from the percentage that would be found by interviewing all undergraduates. YOUR TURN. Find the confidence interval in Exercise 13 “Tennis Challenges.” Bootstrap Resampling for Constructing Confidence Intervals Section 7-4 describes the method of bootstrap resampling for constructing a confidence interval estimate of a population parameter. The basic approach is to use technology such as Statdisk to “resample” the sample data many times (such as 1000), then use the sorted list of 1000 results to find the confidence interval. If we repeat Example 2 using the bootstrap resampling method, here is a typical result: 0.498 6 p 6 0.561. Because of the randomness used in the procedure, the resulting confidence interval may differ somewhat. Analyzing Polls Example 2 deals with a typical poll. When analyzing results from polls, consider the following: 1. The sample should be a simple random sample, not an inappropriate sample (such as a voluntary response sample). 2. The confidence level should be provided. (It is often 95%, but media reports usually fail to identify the confidence level.) 3. The sample size should be provided. (It is often provided by the media, but not always.)
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