Simple Linear Regression (Chapter 14)



Simple Linear Regression – some review exercises

1. The following data represents the years in practice and the annual income in thousand dollars for a random sample of insurance sales representatives. We wish to use regression to predict annual income from experience.

| | |

|YEARS |INCOME |

|X |Y |

| | |

|3 |40 |

| | |

|6 |60 |

| | |

|12 |60 |

| | |

|26 |140 |

| | |

|13 |100 |

Calculate the following values:

a) The slope of the regression line.

b) The standard deviation of Y about the regression line ([pic]).

c) The standard error of the estimate of (1.

d) The standard error for the annual income of a sales representative with 10 years of experience.

2. The following data were collected regarding the monthly starting salaries (in thousand dollars) and the grade point averages (GPA) for undergraduate students who received a degree in business administration:

| | |

|GPA |Monthly |

| |Salary |

| | |

|2.6 |$3.2 |

|3.4 |3.8 |

|3.6 |4.6 |

|3.2 |3.6 |

|3.5 |4.2 |

|2.9 |3.4 |

(a) Develop a scatter diagram for these data with GPA as the independent variable (on the horizontal axis).

(b) What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?

(c) Use the method of least squares to find the estimated regression equation to predict starting salary from GPA.

(d) Use (c) to predict the monthly starting salary for a student with a GPA of 3.1.

(e) Explain what the coefficient of X in the regression equation tells us.

(f) Find the standard error of Y when X = 3.0.

(g) Find the 95% confidence interval of Y when X=3.0

(h) Find the standard error of the mean of Y when X = 3.0

(i) Find the 95% confidence interval of the mean of Y when X = 3.0

(j) Estimate the standard deviation of the regression line's slope.

(k) Find an interval within which you can be 95% confident that b1, the population regression line's slope, will lie.

(l) Compute SSE (sum of squares error, residual).

(m) Compute SST (total sum of squares).

(n) Compute SSR (sum of squares explained by regression).

(o) Compute the coefficient of determination.

(p) explain what your answer in part (o) means.

(q) Use the F-test to check the appropriateness of linear regression.

(r) Compute the sample correlation coefficient between monthly salary and GPA.

(s) Use the t-test to show that linear regression is appropriate.

3. The PJH&D Company is in the process of deciding whether or not to purchase a maintenance contract for its new word processing system. They feel that maintenance expense should be related to usage and have collected the following information on weekly usage (hours) and maintenance expenses.

| | |

|Weekly Usage |Annual Maintenance |

|(Hours) |Expenses ($100's) |

| | |

|13 |17.0 |

|10 |22.0 |

|20 |30.0 |

|28 |37.0 |

|32 |47.0 |

|17 |30.5 |

|24 |32.5 |

|31 |39.0 |

|40 |51.5 |

|38 |40.0 |

(a) Find the estimated regression equation to predict annual maintenance expenses from weekly usage.

(b) Test the significance of the relationship in part (a).

(c) PJH&D expects to operate the word processor 30 hours per week. Develop a 95% prediction interval for the company's annual maintenance expenses.

(d) If the maintenance contract costs $3000 per year, would you recommend purchasing it?

4. In an effort to predict the annual sales of its sales representatives Ace Insurance Company believes it would be appropriate to use the number of years of experience as the independent variable in regression. The company randomly selects 10 salesmen and records their annual sales in ten thousand dollars.

| | |

|ANNUAL SALES |YEARS |

|($100,000) |EXPERIENCE |

|8.0 |1 |

|9.7 |3 |

|9.2 |4 |

|10.2 |4 |

|10.3 |6 |

|11.1 |8 |

|11.9 |10 |

|12.3 |10 |

|11.7 |11 |

|13.6 |13 |

a) Find the value of b1 and explain what its value tells us.

b) Predict the annual sales of a sales representative with 5 years experience.

c) Find the 95% confidence interval of the mean sales of sales representatives with 10 years experience.

d) Explain how and by whom the values calculated in (c) might be used.

e) Find the proportion of the variation which is explained by regression.

SIMPLE REGRESSION PROBLEM

Doctors' incomes have been tested to see how they relate to years of experience. Information was collected on eighteen doctors. Their income and years of experience were recorded. [Note the data table goes onto the next page]

Answer each of the following with an explanation of its meaning in the context of the problem.

|1. Investigate the scatter plot and comment on the appropriateness of linear regression. |Years of Experience |Income |

|2. Find the regression equation to predict the: |2 |($10K) |

|(a) Years of experience from incomes earned. |3 |10.2 |

|(b) Incomes earned from years of experience. (use the answer from part(b) for the following.) |3 |14.1 |

|3. Estimate the standard deviation of y. |4 |12.2 |

|4. Find the standard error of the estimate about the |5 |18.6 |

|the regression line. |5 |16.3 |

|5. Find the 95% confidence interval of the income for a doctor who has been working 3 years. |6 |18.1 |

|6. Find the 95% confidence interval of the mean incomes of the doctors that have been working 3 years. |8 |19.0 |

|7. Find the 95% confidence interval estimate of (1. |8 |21.0 |

|8. Find the 95% confidence interval estimate for (0. |10 |19.5 |

|9. Test the hypothesis that (1 = 0, against the alternative that: |10 |21.9 |

|(a) (1 ( 0 |10 |21.5 |

|(b) (1 > 0. |12 |23.0 |

|10. Explain in the context of the problem the meaning of the value of (1. |13 |24.0 |

| |15 |22.8 |

| |17 |27.1 |

| |18 |27.5 |

| |20 |28.0 |

| | |34.5 |

11. Find the expected annual change in income for a doctor and its 95% confidence interval.

12. Find the proportion of the variation in income that is explained by experience.

13. Find the proportion of the variation in incomes which is not explained by the experience.

14. Concerning the confidence interval of incomes, discuss the interval that would be of concern to the doctor vs. the interval that would be of concern to the insurance company.

15. Discuss the value of this regression equation in predicting the incomes for doctors that have been working 30 years.

16. Discuss the statement, "since the value of R2 is very close to one, we may conclude that years of experience determines income".

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download