A random sample of 65 high school seniors was selected from all high school seniors at a certain high school. The following scatterplot shows the height, in centimeters (cm) , and the foot length, in cm , for each high school senior from the sample. The least-squares regression line is shown. The computer output from the least-squares regression analysis is also shown.
The figure presents a scatterplot in a coordinate plane. The horizontal axis is labeled Foot Length, in centimeters, and the numbers 18 through 34, in increments of 2, are indicated. The vertical axis is labeled Height, in centimeters, and the numbers 150 through 190, in increments of 10, are indicated. There are 65 data points in the scatterplot, and a trend line is given as follows. Note that all values are approximate. The data points begin in the lower left part of the plane at the point with coordinates 19 comma 160. The data points trend upward and to the right and end with the coordinates 33 comma 190. The least-squares regression line begins at the point 18 comma 153, and slants upward and to the right at a constant rate to end at the point 35 comma 197.
Term Coef (SE) Coef T -Value P -Value
Constant 105.08 6.00 17.51 0.000
Foot length 2.599 0.238 10.92 0.000
S=5.90181 R–sq=65.42%
(a) Calculate and interpret the residual for the high school senior with a foot length of 20cm and a height of 160cm .
(b) The standard deviation of the residuals is s=5.9 . Interpret the value in context.
Unit 2 Progress Check: FRQ
Aubree Flores
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Question 1
Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations.
A random sample of 65 high school seniors was selected from all high school seniors at a certain high school. The following scatterplot shows the height, in centimeters (cm), and the foot length, in cm, for each high school senior from the sample. The least-squares regression line is shown. The computer output from the least-squares regression analysis is also shown.
The figure presents a scatterplot in a coordinate plane. The horizontal axis is labeled Foot Length, in centimeters, and the numbers 18 through 34, in increments of 2, are indicated. The vertical axis is labeled Height, in centimeters, and the numbers 150 through 190, in increments of 10, are indicated. There are 65 data points in the scatterplot, and a trend line is given as follows. Note that all values are approximate. The data points begin in the lower left part of the plane at the point with coordinates 19 comma 160. The data points trend upward and to the right and end with the coordinates 33 comma 190. The least-squares regression line begins at the point 18 comma 153, and slants upward and to the right at a constant rate to end at the point 35 comma 197.
Term Coef (SE) Coef T-Value P-Value
Constant 105.08 6.00 17.51 0.000
Foot length 2.599 0.238 10.92 0.000
S=5.90181 R–sq=65.42%
(a) Calculate and interpret the residual for the high school senior with a foot length of 20cm and a height of 160cm.
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Question 2
(b) The standard deviation of the residuals is s=5.9. Interpret the value in context.
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Question 3
(c) The following histogram summarizes the 65 residuals.
The figure presents a histogram. The horizontal axis is labeled Residual, and the numbers negative 20 through 20, in increments of 10, are indicated. The vertical axis is labeled Frequency, and the numbers 0 through 25, in increments of 5, are indicated. The data represented by the bars are as follows. Note that all values are approximate. Residual, negative 20. Frequency, 0. Residual, negative 15. Frequency, 1. Residual, negative 10. Frequency, 5. Residual, negative 5. Frequency, 15. Residual, 0. Frequency, 20. Residual, 5. Frequency, 18. Residual, 10. Frequency, 5. Residual, 15. Frequency, 1. Residual, 20. Frequency, 0.
Assume that the distribution of residuals is approximately normal with mean 0cm and standard deviation 5.9cm. What percent of the residuals are greater than 8cm? Justify your answer.