Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Researchers designed a survey to compare the proportions of children who come to school without eating breakfast in two low-income elementary schools. An SRS of 80 students from School 1 found that 19 had not eaten breakfast, At School 2, an SRS of 150 students included 26 who had not had breakfast. More than 1500 students attend each school. Do these data give convincing evidence of a difference in the population proportions? Carry out a significance test at the a = 0.05 to support your answer. Testing for a Difference in Proportions, p1 – p2 Suppose we have two independent binomial experiments. We would like to test if the two population proportions are equal. Binomial Experiment 1 Binomial Experiment 2 n1 = number of trials n2 = number of trials x1 = number of successes x2 = number of successes p1 = population probability of success on a single trial p2 = population probability of success on a single trial Testing for a Difference in Proportions For large values of n1 and n2 r1 r2 p1 p 2 n1 n 2 Is closely approximated by a normal distribution with p1 p2 p1q1 p2 q 2 n1 n2 where q 1 1 p1 and q 2 1 p2 Testing for a Difference in Proportions The test statistic is as follows: z p1 p2 pc 1 p c n1 p 1 p x1 x 2 where pc n1 n 2 c c n2 How to test a difference of proportions p1 – p2 Consider two independent binomial experiments Binomial Experiment 1 Binomial Experiment 2 n1 = number of trials n2 = number of trials x1 = number of successes out of n1 trials x2 = number of successes out of n2 trials x1 p1 n1 x2 p2 n2 p1 = population probability of success on a single trial p2 = population probability of success on a single trial The Test Procedure 1. Use the null hypothesis of no difference, H0: r1 = r2. Choose the alternate hypothesis in the context of the problem. Set the level of significance a. 2. The null hypothesis is of no difference r1 = r2; therefore pooled (ie., combined) best estimates for the population probabilities of success and failure are x1 x 2 pc and q c 1 pc n1 n 2 The Test Procedure The number of trials should be sufficiently large so that all four quantities n1pc , n1qc , n2pc , n2qc are each larger than 10. Compute the sample test statistic z p1 p2 pc qc pc qc n1 n2 The Test Procedure 3. Use the standard normal distribution and type of test, one-tailed or two-tailed, to find the P-value corresponding to the test statistic. 4. Conclude the test. If P-value < a, then reject H0. If Pvalue > a, then fail to reject the null. 5. State your conclusion in the context of the problem. Researchers designed a survey to compare the proportions of children who come to school without eating breakfast in two low-income elementary schools. An SRS of 80 students from School 1 found that 19 had not eaten breakfast, At School 2, an SRS of 150 students included 26 who had not had breakfast. More than 1500 students attend each school. Do these data give convincing evidence of a difference in the population proportions? Carry out a significance test at the a = 0.05 to support your answer.