Section 10.5

Exercise Set E: 1, 2, 3

Also:

1. The file http://dananne.org/fu/courses/math-241/R/pearson.txt contains the data from Karl Pearson's study of the heights of 1078 fathers and sons discussed in Section 8.1. The data is in a table, with the first column labeled “Father” and the second column labeled “Son”. Recall: If you use RStudio to directly import this file, change the “Separator” selection from “Tab” to “Whitespace”.
   1. Use this data to create a scatter diagram with regression line, treating the heights of the sons as the dependent variable.
   2. Find a 95% prediction interval for the predicted height of a son whose father is 65 inches tall.
   3. Find a 95% confidence interval for the average height of sons whose fathers are 65 inches tall.
   4. Test the hypotheses $H_0 : \beta = 0$ against the hypothesis $H_1 : \beta \ne 0$, where $\beta$ is the true slope of the regression line.
   5. Now use this data to create a scatter diagram with regression line, treating the heights of the fathers as the dependent variable.
   6. Find a 95% prediction interval for the predicted height of a father whose son is 65 inches tall.
   7. Find a 95% confidence interval for the average height of fathers whose sons are 65 inches tall.
2. The file http://dananne.org/fu/courses/math-241/R/reading.txt contains data from a reading test administered to 40 elementary school students. The test was first given to the students in 1982 and then given a year later to the same students. The data is in a table, with the first column labeled “Y1982” and the second column labeled “Y1983”.
   1. Use this data to create a scatter diagram with regression line, treating the 1982 data as the x variable and the 1983 data as the y variable.
   2. Find a 95% prediction interval for the predicted score of a student whose 1982 score was 280.
   3. Find a 95% confidence interval for the average score of students whose 1982 score was 280.
   4. Test the hypotheses $H_0 : \beta = 0$ against the hypothesis $H_1 : \beta \ne 0$, where $\beta$ is the true slope of the regression line.
   5. Use this data to create a scatter diagram with regression line, treating the 1983 data as the x variable and the 1982 data as the y variable.

Answers:

1. 1. 
   2. $(62.5, 72.1)$
   3. $(67.1, 67.5)$
   4. $t = 19.01$, so the p-value is less than $2 \times 10^{-16}$
   5. 
   6. $(61.2, 70.6)$
   7. $(65.7, 66.1)$
2. THe
   1. 
   2. $(236.8, 326.7)$
   3. $(274.5, 289.0)$
   4. $t = 8.774$, so the p-value is $1.14 \times 10^{-10}$
   5.