46. Example 1.2 describes a study of children s private speech (talking to themselves). The 33 children were each observed in about 100 ten-second intervals in the rst grade, and again in the second and third grades. Because private speech occurs more in challenging circumstances, the children were observed while doing their mathematics. The speech was classi ed as on task (about the math lesson), off task, or mumbling (the observer could not tell what was said). Here are the 33 rst-grade mumble scores:
20.8 24.4 19.4 33.3 26.0 56.6 39.5 24.7 21.6 32.1 48.1 19.5 19.2 43.0 26.3 22.7 49.4 35.4 56.8 45.4 28.7 42.2 20.3 20.0 34.0 26.9 48.4 27.6 52.6 5.9 38.5 22.1 22.2 and here are the third-grade mumble scores:
28.8 57.0 23.9 46.9 50.0 64.6 54.2 55.3 21.4 38.3 78.5 38.1 44.3 11.7 58.6 76.1 76.4 48.6 37.2 69.8 29.1 60.4 57.8 38.7 46.5 50.0 69.6 69.8 59.4 22.7 84.9 42.0 67.2 The numbers are in the same order for each grade;
for example, the third student mumbled in 19.4% of the intervals in the rst grade and 23.9% of the intervals in the third grade.

a. Verify graphically that normality is plausible for the population distribution of differences.

b. Find a 95% con dence interval for the difference of population means, and interpret the result.