12.2 Hypothesis Test for One Population Proportion Hypothesis tests for a population proportion. ̂ √ Approximate the normal distribution. So for large sample hypothesis with null hypothesis Ho: p = p o use ̂ √ As test statistic. Called the one-proportion z-test. Example: Economic Stimulus. Gallup conducted a poll to survey 1053 US adults whether they favor an economic stimulus plan of 800 billion dollars. Of those sampled 548 favored passage. At the 5% significance level, do the data provide sufficient evidence to conclude that a majority (more than 50%) of US adults favored passage?
9 Solution: Because n = 1053 and p o = .5 (i.e. 50%) have np o = 1053* .50 = 526.5. and n(1-p o ) = 1053 * (1-.50) = 526.5
10 Step 1: State the null and alternative hypotheses. Let p denote the proportion of all US adults who favored passage of the economic stimulus package. Ho: p = .50 (not true that a majority favored passage) Ha: p > .50 (.50 majority favored) Hypothesis is right-tailed. Step 2: Decide the significance level or . Alpha is .05 Step 3: Compute the value of the test statistic ̂ √ √ ⁄ = 1.30 Step 4: Critical Value Approach Use Table for z right-tailed test. For critical value = 1.645.
11 Step 5 : If the value of the test statistic falls in the rejection region, reject Ho. Otherwise, do not reject. From step 3, the value of the test statistic is z = 1.30, which does not fall in the rejection region. Thus, do not reject Ho. The test results are not statistically significant at the 5% level. P-value approach Step 4: Use Table II to obtain the P-value z = 1.30. The test is right-tailed, so the P-value is the probability of observing a value of z of 1.30 or greater if Ho is true. The probability equals the shaded area, which according to Table II is .0968. Step 5: If P < or = reject Ho. Otherwise, do not reject Ho. Because the P-value of .0968 exceeds the alpha level of .05, do not reject Ho. The data do not provide sufficient evidence to support the null hypothesis. Step 6: Interpret the results of the hypothesis test. At the 5% significance level, the data do not provide sufficient evidence to conclude that a majority of US adults favored passage of the economic stimulus package.
12 12.3 Inferences for Two Population Proportions Compare the proportion of two populations with the same attribute (factor, determinant, predictor) Example: Eating Out Vegetarian Zogby International surveyed 1181 US adults (747 men and 414 women) to ascertain the demand for vegetarian meals in restaurants. Of the 1181 sampled, 276 men and 195 women said they sometimes order vegetarian (no meat, fish or poultry). Research question: Is the percentage of men who sometimes order vegetarian smaller than the percentage of women? a. Formulate the hypothesis. b. Explain how to execute the hypothesis test. c. Explain the relationship between the hypothesis test and decision making (i.e. interpreting results).
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- Spring '14
- Statistics, Statistical hypothesis testing