Consider the following sample data for the relationship between advertising budget and sales for Product A:

Observation

1

2

3

4

5

6

7

8

9

10

Advertising ($)

100,000

110,000

110,000

120,000

130,000

130,000

140,000

150,000

150,000

160,000

Sales ($)

603,000

676,000

655,000

748,000

796,000

785,000

858,000

891,000

935,000

980,000

**What is the slope of the "least-squares" best-fit regression line?**

Please round your answer to the nearest hundredth.

To calculate the slope of the least-squares best-fit regression line, we use the formula for the slope $ m $ in simple linear regression:

\[
m = \frac{n \sum{(x_i y_i)} - \sum{x_i} \sum{y_i}}{n \sum{(x_i^2)} - (\sum{x_i})^2}
\]

Where:

- $ x_i $ represents the advertising budget
- $ y_i $ represents the sales
- $ n $ is the number of observations (in this case, $ n = 10 $)

We will now compute this step by step.

### Step 1: Calculate the necessary sums:
- $ \sum{x_i} $: Sum of advertising budgets
- $ \sum{y_i} $: Sum of sales
- $ \sum{(x_i^2)} $: Sum of squares of advertising budgets
- $ \sum{(x_i y_i)} $: Sum of the product of advertising budgets and sales

Let’s calculate these values.The necessary sums are:

- $ \sum{x_i} = 1,300,000 $
- $ \sum{y_i} = 7,927,000 $
- $ \sum{x_i^2} = 172,600,000,000 $
- $ \sum{x_i y_i} = 1,052,820,000,000 $

### Step 2: Substitute these values into the slope formula:

\[
m = \frac{n \sum{(x_i y_i)} - \sum{x_i} \sum{y_i}}{n \sum{(x_i^2)} - (\sum{x_i})^2}
\]

Now, let's compute the slope $ m $.The slope of the least-squares best-fit regression line is $ 6.2 $, rounded to the nearest hundredth.

Would you like more details or have any further questions? Here are five related questions that build on this concept:

1. How do we interpret the slope in this context?
2. What is the equation of the least-squares regression line?
3. How can we use this model to predict sales for a specific advertising budget?
4. What is the y-intercept of the least-squares regression line?
5. How do we compute the coefficient of determination (R²) for this regression?

**Tip:** The slope of a regression line represents the change in the dependent variable (sales) for every unit increase in the independent variable (advertising budget).

What is the slope of the 'least-squares' best-fit regression line given the relationship between advertising budget and sales for Product A?

Find the slope of the least-squares regression line based on Product A's advertising budget and sales data. This problem covers key statistical concepts in linear regression, using the least-squares method to determine the best-fit line. Suitable for advanced high school or college-level students.

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