Question

Find the equation for the least squares regression line of the data described below.
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
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the equation of the least squares regression line (also known as the line of best fit) for the given data, we need to perform several steps including calculating the necessary statistics: means, sums of squares, and slopes.

### Step 1: Data Preparation
Let's summarize the data you provided:

| x (movie duration) | y (% customers buying snacks) |
|---------------------|-------------------------------|
| 103 | 5.16 |
| 116 | 13.35 |
| 118 | 4.90 |
| 139 | 19.95 |
| 149 | 24.29 |

### Step 2: Calculate the Means
First, we calculate the mean of \(x\) and \(y\).

\[
\overline{x} = \frac{103 + 116 + 118 + 139 + 149}{5} = \frac{625}{5} = 125
\]

\[
\overline{y} = \frac{5.16 + 13.35 + 4.90 + 19.95 + 24.29}{5} = \frac{67.65}{5} = 13.53
\]

### Step 3: Calculate the Slope (b) of the Regression Line
The formula for the slope \(b\) of the least squares regression line is:

\[
b = \frac{\sum (x_i - \overline{x})(y_i - \overline{y})}{\sum (x_i - \overline{x})^2}
\]

### Step 4: Calculate the Required Summations
Now we calculate the necessary components.

1. **Compute \( \sum (x_i - \overline{x})(y_i - \overline{y}) \)**

| \(x_i\) | \(y_i\) | \(x_i - \overline{x}\) | \(y_i - \overline{y}\) | \((x_i - \overline{x})(y_i - \overline{y})\) |
|----------|----------|----------------|----------------|---------------------------|
| 103 | 5.16 | -22 | -8.37 | 183.14 |
| 116 | 13.35 | -9 | -0.18 | 1.62 |
| 118 | 4.90 | -7 | -8.63 | 60.41 |
| 139 | 19.95 | 14 | 6.42 | 89.88 |
| 149 | 24.29 | 24 | 10.76 | 258.24 |

\[
\sum (x_i - \overline{x})(y_i - \overline{y}) = 183.14 + 1.62 + 60.41 + 89.88 + 258.24 = 593.29
\]

2. **Compute \( \sum (x_i - \overline{x})^2 \)**

\[
| \overline{x} - x_i|^2
\]

| \(x_i\) | \(x_i - \overline{x}\) | \((x_i - \overline{x})^2\) |
|----------|----------------|-----------------------|
| 103 | -22 | 484 |
| 116 | -9 | 81 |
| 118 | -7 | 49 |
| 139 | 14 | 196 |
| 149 | 24 | 576 |

\[
\sum (x_i - \overline{x})^2 = 484 + 81 + 49 + 196 + 576 = 1386
\]

### Step 5: Calculate the Slope (b)
Now we can calculate \(b\):

\[
b = \frac{593.29}{1386} \approx 0.428
\]

### Step 6: Calculate the Intercept (a)
Using the formula for the intercept \(a\):

\[
a = \overline{y} - b \cdot \overline{x}
\]

\[
a = 13.53 - 0.428 \cdot 125
\]
\[
a = 13.53 - 53.5 \approx -39.97
\]

### Step 7: Write the Equation
The equation for the least squares regression line is:

\[
y = bx + a
\]

Plugging in for \(b\) and \(a\):

\[
y \approx 0.428x - 39.970
\]

### Final Result
Rounding to the nearest thousandth:

\[
y = 0.428x - 39.970
\]

This is the least squares regression line representing the relationship between movie duration and the percentage of customers who buy snacks.