Question

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In December 2004, 39% of students in high school were satisfied with the lunches supplied through the school. In May 2010, an organization conducted a poll of 851 students in high school and asked if they were satisfied with the lunches supplied through the school of the 851 surveyed, 298 indicated they were satisfied. Does thi suggest the proportion of students satisfied with the quality of lunches has decreased? (a) What does it mean to make a Type Il error for this test? (b) if the researcher decides to test this hypothesis at the a=0.10 level of significance, compute the probability of making a Type Il error, B. if the true population proportion is 0.35. What is the power of the test? (c) Redo part (b) if the true population proportion is 0.38. (a) What does it mean to make a Type Il error for this test? Choose the correct answer below. O A. Ho is rejected and the true proportion of high school students who are not satisfied with the quality of lunches is less than 0.39. O B . Hy is not rejected and the true proportion of high school students who are not satisfied with the quality of lunches is equal to 0.39. O C. Ho is not rejected and the true proportion of high school students who are not satisfied with the quality of lunches is less than 0.39. (b) If the researcher decides to test this hypothesis at the a=0.10 level of significance, compute the probability of making a Type Il error, B, if the true population proportion is 0.35. What is the power of the test? B=0 Power = (Type integers or decimals rounded to four decimal places as needed.) (c) Redo part (b) if the true population proportion is 0.38. BE Power = (Type integers or decimals rounded to four decimal places as needed.)

Answer 1

(a) The Type I error: The probability of Type II error is the probability of failing to reject the null hypothesis H., when the null hypothesis (Ho) is false. In the context of the problem, the probability of not rejecting the false null hypothesis is that the true proportion of high school students who are not satisfied with the quality of the lunches is equal to 0.39. - - - - - - - - - - - - - - - - - (6) The sample size is n = 851. The true population proportion is p = 0.35. The population proportion is P, = 0.39. The formula for calculating the Type I error of the test is (Po-P)-2. [Po(1,5 Po) 8=1-0 P(1-P) (0.39-0.35)-2010 V 10.39(1-0.39) 1-0 851 0.35(1-0.35) V 851 0.39(0.61) fo35(1-035) =1-0 (0.04)– 20.104 851 10.35(0.65) 10.35(0.65) V 851 =1-0 (0.04)– 20.10 V0.00028 V0.000267 ((0.04)– 1.281552(0.01672) =1-01 – 0.01635 (0.04)-0.021427) =1- 0.01635 ) 1-0 0.018573 - 0.01635) =1-0(1.13592) = 1-0.872005 0.127995 B0.1279 Therefore, the probability of Type I error is $ 0.1279 - -------- ------------ -- ------ The formula for calcualting the power of the test is - - - - - - - - - - - - - - - 1-8 1-0.1279 =0.872005 Therefore, the power of the test is 1-B 10.8720) - - - - - - - -

(C) The sample size is n = 851. The true population proportion is p = 0.35. The population proportion is po = 0.38. The formula for calculating the Type I error of the test is Di P. (1-P) 8=1-0 (Po-p)-27 (1-P) (0.39-0.38) – 20.10 9 851 (0.04)– 20.104 851 10.39(1-0.39) 1-0 - 10.38(1-0.38) V 851 0.39(0.61) 1- 0.2356 V 851 ((0.04) – 20.10 V0.00028 1-0 V0.000277 1-0 (0.04)– 1.281552 (0.01672) 0.016639 1-0 (0.04)-0.021427 0.016639 >1- 0.018573 10.016639) =1-0(1.116223) = 1-0.867837 0.132163 B$ 0.132163 Therefore, the probability of Type I error is B 0.1322 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - The power of the test is 1-8 1-0.132163 =0.867837 Therefore, the power of the test is 1-0 0.8678