426 CHAPTER 8 Hypothesis Testing In this section we include these resampling methods: ■ Bootstrap The bootstrap resampling method is used to construct a confidence interval that could be used to test a claim about a population parameter. See Section 7-4 for more detail on bootstrap resampling. ■ Randomization Randomization, which is used for hypothesis testing, is another method of resampling. While bootstrapping generates resamples of the same original data, randomization requires that the sample data be modified to agree with the assumption used in the null hypothesis. The bootstrap resampling method was first introduced in Section 7-4. We now introduce the randomization method of resampling. Randomization applies to tests of claims about a population proportion, population mean, or population standard deviation. Randomization DEFINITION Randomization is a method of hypothesis testing that involves resampling (with replacement) after the sample data have been modified to conform to the value of the population parameter that is assumed in the null hypothesis. Resample with Replacement? When using randomization for a hypothesis test of a claim about one proportion, mean, or standard deviation, resampling from one sample without replacement would result in the same generated sample every time, which is not useful. Instead, when working with one sample, we resample with replacement. POOR TERMINOLOGY Because we have already used the terms of “random” and “randomness” to describe sampling methods in Chapter 1, it is poor terminology to use “randomization” for a method of resampling, but that is the term now in use. Another term for a randomization test is a “permutation test,” but that is also poor terminology because combinations are used instead of permutations. A good term for randomization would be “resampling for hypothesis tests,” but apparently that is not snappy enough. The following example illustrates randomization with one sample. Randomization Method EXAMPLE 1 Assume that we want to use randomization to test the claim that the mean service time (minutes) at Mario’s Pizza food truck is equal to 6.5 minutes. That claim results in the following hypotheses H0: m = 6.5 minutes H1: m ≠ 6.5 minutes Shown on the next page in the leftmost column is this sample of five wait times (minutes): 2, 3, 6, 8, 11. Because this sample has a mean of x = 6.0 minutes but the hypothesis test is conducted with the assumption that m = 6.5 minutes, we modify the original data set by adding 0.5 minute to each value so that the mean becomes x = 6.5 minutes as assumed. We then resample the modified data, and one such resampling is shown on the next page in the rightmost column.
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