image 7.47 Assessment of a foreign-language institute. 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 four weeks. At the beginning of the period, the teachers were given the Modern Language Association’s listening test of understanding of spoken French. After four 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.) The maximum possible score on the test is 36.21 Here are the data:

Teacher Pretest Posttest Gain Teacher Pretest Posttest Gain
1 32 34 2 11 30 36 6
2 31 31 0 12 20 26 6
3 29 35 6 13 24 27 3
4 10 16 6 14 24 24 0
5 30 33 3 15 31 32 1
6 33 36 3 16 30 31 1
7 22 24 2 17 15 15 0
8 25 28 3 18 32 34 2
9 32 26 −6 19 23 26 3
10 20 26 6 20 23 26 3

To analyze these data, we first subtract the pretest score from the posttest score to obtain the improvement for each teacher. These 20 differences form a single sample. They appear in the “Gain” columns. The first teacher, for example, improved from 32 to 34, so the gain is 34 − 32 = 2.

  1. (a) State appropriate null and alternative hypotheses for examining the question of whether or not the course improves French spoken-language skills.

  2. (b) Describe the gain data. Use numerical and graphical summaries.

  3. (c) Perform the significance test. Give the test statistic, the degrees of freedom, and the P-value. Summarize your conclusion.

  4. (d) Give a 95% confidence interval for the mean improvement.