A random sample of 65 high school seniors was selected from all high school seniors at a certain high school. The following scat

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2 months ago

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A random sample of 65 high school seniors was selected from all high school seniors at a certain high school. The following scat

terplot 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.
(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.
(c) The following histogram summarizes the 65 residuals.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.
(d) Based on your answer to part (c), would it be surprising to randomly select a high school senior from the high school with a foot length of 20cm and a height greater than 165cm ? Justify your answer.

Mathematics

1 answer:

Svet_ta [12.7K] · Answer 1 · 2026-02-17T21:25:00+03:00

Step-by-step explanation:

a) Utilizing the least-squares regression equation, the predicted height for a senior with a foot length of 20 cm is:

y = 2.599x + 105.08

y = 2.599(20) + 105.08

y = 157.06

The residual is 160 − 157.06 = 2.94

b) The variability between each data point and its estimate has a standard deviation of 5.9 cm.

c) To find the z-score:

z = (x − μ) / σ

z = (8 − 0) / 5.9

z = 1.36

Use a table or calculator to compute the probability.

P(Z > 1.36) = 1 − 0.9131 = 0.0869

Hence, around 8.69% of the residuals exceed 8 cm.

d) As the z-score lies within ±2 standard deviations, it's not unexpected to randomly choose a senior with a foot length of 20 cm and a height of 165 cm.