University:
Missouri State University
Course:
MTH 343 | Foundations of Probability and Statistics for Teachers
Academic year:

2021
Views:

416

Pages:

7
Author:

Kevin Riddle

STATISTICS SCORING GUIDELINES Question 4 Intent of Question The primary goal of this question was to assees students ability to identify, set up, perform, and interpret the results of an appropriate hypothesis test to address a particular question. More specific goals were to assess students' ability to (1) state appropriate hypotheses, (2) identify the name of an appropriate statistical test and check appropriate assumptions/conditions, (3) calculate the appropriate test statistic and p-value (4) draw an appropriate conclusion, with justification, in the context of the study Solution Step 1: States a correct pair of hypotheses. Let pay represent the population proportion of adults in the United States who would have answered "yes" about the effectiveness of television commerciais in December 2007. Let Pon represent the analogous population proportion in December 2008 The hypotheses to be tested are Ho Por Pos versus Ha Por Pos Step 2: Identifies a correct test procedure (by name or by formula) and checks appropriate conditions The appropriate procedure is a two-sample z-test for comparing proportions. Because these are sample surveys, the first condition is that the data were gathered from independent random samples from the two populations. This condition is met because we are told that the subjects were randomly selected in the two different years. Although we are not told whether the samples were selected independently, this is a reasonable assumption given that they are samples of different sizes selected in different years The second condition is that the sample sizes are large, relative to the proportions involved. This condition is satisfied because all sample counts (622 "yes" in 2007, 1,020-622398 "no" in 2007 676 "yes" in 2008; 1,009-676333 "no" in 2008) are all at least 10 (or, are all at least 5). An additional condition may be checked: The population sizes (more than 200 million adults in the United States) are much larger than 10 (or, 20) times the sample sizes Step 3. Correct mechanics, including the value of the test statistic and p-value (or rejection region). The sample proportions who answered "yes" are Pen 622 1,020 0.6098 and hat p_{99} = 676/1009 = 0.67 . The combined proportion. p. who answered "yes" in these two years is: hat p_{c} = (622 + 676)/(1020 + 1.009) = (1, 298)/2029 = 0.6397 . The test statistic is tau= hat P 12 - hat P 39 sqrt hat p x (1- hat P x ) (1/R_{7} + 1/R_{7}) = 0.60999-0.6700 sqrt (0.6397X1-0.6397( 1 1.020 + 1 1.000 ) =-2.82 The p-value is 2P(Z <= - 2.82) = 0.0048 . Step 4: State a correct conclusion in the context of the study, using the result of the statistical test. Because this p-value is smaller than any common significance level such as a 0.05 of a = 0.01 (or, because this p-value is so small), we reject Ho and conclude that the data provide convincing. (or, statistically significant) evidence that the proportion of all adults in the United States who would answer "yes" to the question about the effectiveness of television commercials changed. from December 2007 to December 2008. Reject He