4 A Summary of Forms for Null and Alternative Hypotheses about a Population Proportion The equality part of the hypotheses always appears in the null hypothesis. In general, a hypothesis test about the value of a population proportion p must take one of the following three forms (where p 0 is the hypothesized value of the population proportion). H 0 : p > p 0 H 0 : p < p 0 H 0 : p = p 0 H a : p p 0 H a : p p 0
5 Tests about a Population Proportion: Large-Sample Case (np > 5 and n(1 - p) > 5) Test Statistic where: Rejection Rule One-Tailed Two-Tailed H 0 : p p Reject H 0 if z > z H 0 : p p Reject H 0 if z < -z H 0 : p p Reject H 0 if |z| > z
6 Example: NSC Two-Tailed Test about a Population Proportion: Large n –Hypothesis H 0 : p =.5 H a : p.5 –Test Statistic
7 Contoh Soal: NSC Two-Tailed Test about a Population Proportion: Large n –Rejection Rule Reject H 0 if z 1.96 –Conclusion Do not reject H 0. For z = 1.278, the p-value is.201. If we reject H 0, we exceed the maximum allowed risk of committing a Type I error (p-value >.050).
8 Tests of Goodness of Fit and Independence Goodness of Fit Test: A Multinomial Population Tests of Independence: Contingency Tables Goodness of Fit Test: Poisson and Normal Distributions
9 Goodness of Fit Test: A Multinomial Population 1. Set up the null and alternative hypotheses. 2. Select a random sample and record the observed frequency, f i, for each of the k categories. 3. Assuming H 0 is true, compute the expected frequency, e i, in each category by multiplying the category probability by the sample size. continued
10 Goodness of Fit Test: A Multinomial Population 4. Compute the value of the test statistic. 5. Reject H 0 if (where is the significance level and there are k - 1 degrees of freedom).
11 Contoh Soal: Finger Lakes Homes Multinomial Distribution Goodness of Fit Test The number of homes sold of each model for 100 sales over the past two years is shown below. Model Colonial Ranch Split-Level A- Frame # Sold 30 20 35 15