2. The National Endowment for the Humanities sponsors summer institutes to improve the skills of high school teachers of foreign languages. One such institute hosted 20 French teachers for 4 weeks. At the beginning of the period the teachers were given the Modern Language Association's listening test of understanding of spoken French. After 4 weeks of immersion in French in and out of class, the listening test was given again. (The actual French spoken in the two tests was different, so that simply taking the first test should not improve the score on the second test.) To analyze these data, we first subtract the pretest score from the posttest score to obtain improvement for each teacher. These 20 differences form a single sample and are summarized under the heading "Gain': Pre Post Gain mean 25.8 28.3 2.50 standard deviation 6.31 5.95 2.89 Answers may be rounded, but use 4 or more significant digits in your calculations. (a) [1 pt] State the appropriate null, Ho, and alternative, Ha, hypotheses for examining the question of whether o not the course improves French spoken-language skills. Use the parameter u (mu). (b) [4 pts] Perform the significance test. Give the test statistic, the degrees of freedom, and the p-value. Summarize your conclusion. (c) [4 pts] Construct the 96% and 98% confidence interval for mean improvement. For each of the following, answer simply yes or no using your results from (c): (d) [1/2 pt] Do these data reject Ho : u = 1 against Ha : u / 1 at the a = .02 level? (e) [1/2 pt] Do these data reject Ho : u = 1 against Ha : u / 1 at the or =.04 level? (f) [1/2 pt] Do these data reject Ho : u = 1 against H. : / > 1 at the or = .04 level? g) [1/2 pt] Do these data reject H. : u = 1 against Ha : u > 1 at the a = .02 level? MacBook Pro