Question

Use the least squares regression line of this data set to predict a value.
A movie theater manager is studying the snack buying habits of her customers. She expects that when longer movies are shown, people buy more snacks. Each week she calculates the average duration of all movies being shown and the percentage of customers who buy snacks.
The manager calculates the average movie duration (in minutes), x, and the percentage of the customers who bought snacks, y.
Average movie duration Percentage of customers that bought snacks
103 5.16
116 13.35
118 4.90
139 19.95
149 24.29
The least squares regression line of this data set is:
y=0.429x–40.068
If the average movie duration is 138 minutes, what percentage of customers does this line predict will buy snacks?
Round your answer to the nearest thousandth.
percent

Answer 1

To predict the percentage of customers that will buy snacks when the average movie duration is 138 minutes using the least squares regression line given by the equation:

\[
y = 0.429x - 40.068
\]

we can substitute \( x = 138 \) into the equation:

\[
y = 0.429(138) - 40.068
\]

Now, calculating it step-by-step:

1. Calculate \( 0.429 \times 138 \):
\[
0.429 \times 138 = 59.058
\]

2. Subtract \( 40.068 \) from the result:
\[
59.058 - 40.068 = 18.990
\]

Thus, the predicted percentage of customers who buy snacks when the average movie duration is 138 minutes is:

\[
y \approx 18.990
\]

Rounding to the nearest thousandth, the final answer is:

\[
\boxed{18.990} \text{ percent}
\]