8-2 Testing a Claim About a Proportion 391 use of confidence intervals. In addition, Section 8-5 describes the alternative resampling methods of bootstrapping and randomization. The methods of this section can be used with claims about population proportions, probabilities, or the decimal equivalents of percentages. There are different methods for testing a claim about a population proportion. Part 1 of this section is based on the use of a normal approximation to a binomial distribution, and this method serves well as an introduction to basic concepts, but it is not a method used by professional statisticians. Part 2 discusses other methods that might require the use of technology. HINT: FINDING THE NUMBER OF SUCCESSES When using technology for hypothesis tests of proportions, we must usually enter the sample size n and the number of successes x, but in real applications the sample proportion pn is often given instead of x. The number of successes x can be easily found by evaluating x = npn. If 52% of 926 survey respondents answer “yes” to a question, the number who answered “yes” is x = n# pn = 1926210.522 = 481.52, which we must round to x = 482. (481.52 must be rounded to a whole number because it is a count of the number of people among 926, and also because technology typically requires entry of a whole number.) Caution: When conducting hypothesis tests of claims about proportions, slightly different results can be obtained when calculating the test statistic using a given sample proportion instead of using a rounded value of x found by using x = npn. PART 1 Normal Approximation Method The box on the next page includes the key elements used for testing a claim about a population proportion. The following procedure is based on the use of a normal distribution as an approximation to a binomial distribution. When obtaining sample proportions from samples that all have the same size n, and if n is large enough, the distribution of the sample proportions is approximately a normal distribution. Equivalent Methods When testing claims about proportions, the P-value method and the critical value method are equivalent to each other in the sense that they both lead to the same conclusion. However, the confidence interval method is not equivalent to them. (Both the P-value method and the critical value method use the same standard deviation based on the claimed proportion p, but the confidence interval method uses an estimated standard deviation based on the sample proportion.) So the confidence interval method could result in a different conclusion. Recommendation: Use the P-value method or critical value method for testing a claim about a proportion (see Exercise 34), but use a confidence interval to estimate a population proportion.
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