392 CHAPTER 8 Hypothesis Testing Claim: Most Internet Users Utilize Two-Factor Authentication to Protect Their Online Data The Chapter Problem cited a Pew Research Center survey in which 926 Internet users were asked if they utilize two-factor authentication on at least one online account, and 52% of them responded with “yes.” Use this result to test the claim that most Internet users utilize two-factor authentication to protect their online data. REQUIREMENT CHECK We first check the three requirements. 1. The 926 consumers are randomly selected. 2. There is a fixed number (926) of independent trials with two categories (the respondent either utilizes two-factor authentication or does not). CP Testing a Claim About a Population Proportion (Normal Approximation Method) Objective Conduct a formal hypothesis test of a claim about a population proportion p. Notation n = sample size or number of trials p = population proportion (p is the value used in the statement of the null hypothesis) pn = x n (sample proportion) q = 1 - p KEY ELEMENTS Requirements 1. The sample observations are a simple random sample. 2. The conditions for a binomial distribution are satisfied: • There is a fixed number of trials. • The trials are independent. • Each trial has two categories of “success” and “failure.” • The probability of a success remains the same in all trials. 3. The conditions np Ú 5 and nq Ú 5 are both satisfied, so the binomial distribution of sample proportions can be approximated by a normal distribution with m = np and s = 1npq (as described in Section 6-6). Note that p used here is the assumed proportion used in the claim, not the sample proportion pn. (If this requirement is not satisfied, test the claim using a confidence interval obtained by using the resampling methods of bootstrapping, or use randomization described in Section 8-5, or use an exact method described in Part 2 of this section, or use the sign test described in Section 13-2.) Test Statistic for Testing a Claim About a Proportion z = pn - p Apq n P-values: P-values are automatically provided by technology. If technology is not available, use the standard normal distribution (Table A-2) and refer to Figure 8-3 on page 380. Critical values: Use the standard normal distribution (Table A-2).
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