A sales manager collected the following data on *x* = years of experience and *y* = annual sales ($1,000s). The estimated regression equation for these data is

*ŷ* = 81 + 4*x*.

SalespersonYears of
ExperienceAnnual Sales
($1,000s)

1

1

80

2

3

97

3

4

97

4

4

102

5

6

103

6

8

111

7

10

119

8

10

128

9

11

117

10

13

136

(a)

Compute SST, SSR, and SSE.

SST=SSR=SSE=

(b)

Compute the coefficient of determination

*r*2.

(Round your answer to three decimal places.)

*r*2

=

Comment on the goodness of fit. (For purposes of this exercise, consider a proportion large if it is at least 0.55.)

The least squares line provided a good fit as a large proportion of the variability in *y* has been explained by the least squares line.The least squares line provided a good fit as a small proportion of the variability in *y* has been explained by the least squares line.    The least squares line did not provide a good fit as a small proportion of the variability in *y* has been explained by the least squares line.The least squares line did not provide a good fit as a large proportion of the variability in *y* has been explained by the least squares line.

(c)

What is the value of the sample correlation coefficient? (Round your answer to three decimal places.)

Let's break down the given problem and solve each part systematically.

### Part (a) Compute SST, SSR, and SSE.

To calculate **SST** (Total Sum of Squares), **SSR** (Regression Sum of Squares), and **SSE** (Error Sum of Squares), we need to use the following formulas:

1. **SST** (Total Sum of Squares):
   \[
   SST = \sum(y_i - \bar{y})^2
   \]
   where $ y_i $ are the actual values of annual sales and $ \bar{y} $ is the mean of the $ y_i $ values.

2. **SSR** (Regression Sum of Squares):
   \[
   SSR = \sum(\hat{y}_i - \bar{y})^2
   \]
   where $ \hat{y}_i $ are the predicted values of sales from the regression equation $ \hat{y} = 81 + 4x $.

3. **SSE** (Error Sum of Squares):
   \[
   SSE = \sum(y_i - \hat{y}_i)^2
   \]
   where $ y_i $ are the actual sales values and $ \hat{y}_i $ are the predicted sales values.

### Part (b) Compute the coefficient of determination $ r^2 $.

The coefficient of determination $ r^2 $ is calculated using the formula:
\[
r^2 = \frac{SSR}{SST}
\]
$ r^2 $ tells us the proportion of variability in the dependent variable (annual sales) that can be explained by the regression model (years of experience).

### Part (c) Compute the sample correlation coefficient $ r $.

The sample correlation coefficient $ r $ is calculated by taking the square root of $ r^2 $:
\[
r = \sqrt{r^2}
\]

### Steps to solve:

1. We have the data for years of experience ($x$) and actual annual sales ($y$).
2. We'll calculate the predicted sales ($\hat{y} = 81 + 4x$) for each salesperson.
3. Calculate the necessary sums for SST, SSR, and SSE.
4. Compute $ r^2 $ and $ r $.

Let me compute these values for you.### Part (a) SST, SSR, and SSE:

- **SST (Total Sum of Squares):** 2472
- **SSR (Regression Sum of Squares):** 2272
- **SSE (Error Sum of Squares):** 200

### Part (b) Coefficient of Determination ($ r^2 $):

\[
r^2 = 0.919
\]

This means that approximately 91.9% of the variability in the annual sales can be explained by the number of years of experience.

- **Comment on the goodness of fit:** The least squares line provided a good fit as a large proportion of the variability in $ y $ has been explained by the least squares line.

### Part (c) Sample Correlation Coefficient ($ r $):

\[
r = 0.959
\]

This shows a strong positive correlation between years of experience and annual sales.

---

Would you like more details or have any questions?

Here are 5 related questions:
1. How can we interpret the slope of the regression equation?
2. What would the predicted sales be for a salesperson with 7 years of experience?
3. How do we know if the regression model is statistically significant?
4. What other factors could potentially explain the variance in sales besides experience?
5. Can we predict sales accurately for someone with 20 years of experience using this model?

**Tip:** The closer $ r^2 $ is to 1, the better the model explains the variability in the data.

A sales manager collected the following data on x = years of experience and y = annual sales ($1,000s). The estimated regression equation for these data is ŷ = 81 + 4x. (a) Compute SST, SSR, and SSE. (b) Compute the coefficient of determination r². (c) What is the value of the sample correlation coefficient?

This page provides a detailed solution to calculating SST, SSR, SSE, and the coefficient of determination (r²) for a regression analysis involving years of experience and annual sales. Learn the step-by-step process for computing these values, understanding the goodness of fit, and determining the sample correlation coefficient.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation