An advertising firm wishes to demonstrate to its clients the effectiveness of the advertising campaigns it has conducted. The following bivariate data on twelve recent campaigns, including the cost of each campaign (denoted by , in millions of dollars) and the resulting percentage increase in sales (denoted by ) following the campaign, were presented by the firm. A scatter plot of the data is shown in Figure 1.
Campaign cost, x
(in millions of dollars)	Increase in sales, y
(percent)
3.23	6.54
3.47	6.83
2.84	6.88
1.53	6.59
2.50	6.60
1.95	6.71
1.44	6.37
3.80	6.78
1.98	6.58
3.79	6.96
1.52	6.18
2.94	6.68
Increase in sales
(percent)	
y66.26.46.66.877.2x11.522.533.54
Campaign cost
(in millions of dollars)
 Figure 1
The least-squares regression line for these data has a slope of approximately .
Answer the following. Carry your intermediate computations to at least four decimal places, and round your answers as specified below. (If necessary, consult a list of formulas.)
(a)	What is the value of the y-intercept of the least-squares regression line for these data?
Round your answer to at least two decimal places.

(b)	What is the value of the sample correlation coefficient for these data?
Round your answer to at least three decimal places.


To solve the problem, we need to calculate the following:

1. **The y-intercept of the least-squares regression line** (denoted as \( b_0 \)).
2. **The sample correlation coefficient** (denoted as \( r \)).

Given:
- The slope \( b_1 \) is provided by the problem.
- The data points are given for campaign costs \( x \) and the corresponding increase in sales \( y \).

### Step 1: Find the mean values of \( x \) and \( y \).

Let's calculate the means \( \bar{x} \) and \( \bar{y} \) first.

\[
\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i
\]

\[
\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i
\]

where \( n \) is the number of data points.

### Step 2: Calculate the y-intercept \( b_0 \).

The y-intercept \( b_0 \) is calculated using the formula:

\[
b_0 = \bar{y} - b_1 \cdot \bar{x}
\]

### Step 3: Calculate the sample correlation coefficient \( r \).

The sample correlation coefficient \( r \) is calculated using:

\[
r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2 \cdot \sum_{i=1}^{n} (y_i - \bar{y})^2}}
\]

Let me calculate these values step by step.### Results:
(a) The value of the **y-intercept** \( b_0 \) of the least-squares regression line is approximately **6.26** (rounded to two decimal places).

(b) The value of the **sample correlation coefficient** \( r \) for these data is approximately **0.743** (rounded to three decimal places).

Would you like further details on the computations or have any other questions? Here are five related questions you might be interested in:

1. How is the slope of the regression line determined?
2. What does the correlation coefficient tell us about the relationship between campaign cost and sales increase?
3. How would the regression line change if one data point was removed?
4. How do you interpret the y-intercept in this context?
5. What are the assumptions underlying linear regression?

**Tip:** A strong correlation coefficient near 1 or -1 indicates a strong linear relationship, while a coefficient near 0 suggests a weak or no linear relationship.

Explore how an advertising firm evaluates campaign effectiveness using statistical analysis. Learn about calculating the y-intercept of the least-squares regression line and the sample correlation coefficient based on twelve recent campaigns' data. Round your answers to the specified decimal places.

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