Question

Use the least squares regression line of this data set to predict a value.
Researchers at a sleep clinic want to know how watching television affects a person's ability to fall asleep. In a study, they measure the amount of time it takes subjects to fall asleep at night after watching television.
The researchers note the number of minutes the subjects spent watching television in the evening, x, as well as the number of minutes it took them to fall asleep, y.
Minutes of television Minutes to fall asleep
38 16
44 34
89 50
91 36
92 46
The least squares regression line of this data set is:
y=0.394x+8.477
How many minutes does this line predict it will take a subject to fall asleep if they watch 40 minutes of television in the evening?
Round your answer to the nearest integer.
minutes

Answer 1

To predict the number of minutes it will take a subject to fall asleep after watching 40 minutes of television, we will use the least squares regression equation given:

\[ y = 0.394x + 8.477 \]

where \( x \) is the number of minutes spent watching television. We will substitute \( x = 40 \) into the equation:

\[
y = 0.394(40) + 8.477
\]

Calculating \( 0.394 \times 40 \):

\[
0.394 \times 40 = 15.76
\]

Now, add this result to \( 8.477 \):

\[
y = 15.76 + 8.477 = 24.237
\]

Rounding \( 24.237 \) to the nearest integer, we get:

\[
y \approx 24
\]

So, the line predicts it will take approximately **24 minutes** for a subject to fall asleep if they watch 40 minutes of television in the evening.