Problems of Regression

1. A company wants to predict the annual value of its total sales based on the national income of the country where is does business. The relationship is represented in the following table:

x 189 190 208 227 239 252 257 274 293 308 316
y 402 404 412 425 429 436 440 447 458 469 469

x represents the national income in millions of dollars and y represents the company's sales in thousands of dollars in the period from 1990 to 2000 (inclusive). Calculate:

1 The regression line of y on x.

2 The linear correlation coefficient and interpret it.

3 If in 2001, the country's national income was 325 million dollars, what would the prediction for the company's sales be?


2. The statistical information obtained from a sample of 12 farms on the relationship between the investment and yield in hundreds of thousands of dollars is shown in the following table:

Investment (x) 11 14 16 15 16 18 20 21 14 20 19 11
Yield (y) 2 3 5 6 5 3 7 10 6 10 5 6

Calculate:

1 The regression line of the yield with regard to the investment.

2 The estimated investment needed to obtain a yield of $1,250,000.


3. The number of hours devoted to studying a subject and the marks obtained by eight students in the corresponding examination is:

Hours (x) 20 16 34 23 27 32 18 22
Mark (y) 6.5 6 8.5 7 9 9.5 7.5 8

Calculate:

1 Line of regression of y on x.

2 The estimated mark a person would obtain who studied 28 hours.


4. The following table shows the age (in years) of 10 children and a quantitative measure of their aggressive behavior (measured on a scale of 0 to 10)

Age 6 6 6.7 7 7.4 7.9 8 8.2 8.5 8.9
Aggressive behavior 9 6 7 8 7 4 2 3 3 1

1 Determine the regression line of aggressive behavior according to age.

2 From that line, determine the value of aggressive behavior that would correspond to a child of 7.2 years.


5. The values of two variables x and y are distributed according to the following table:

y/x 100 50 25
14 1 1 0
18 2 3 0
22 0 1 2

1 Calculate the covariance.

2 Obtain and interpret the linear correlation coefficient.

3 Determine the equation of the regression line of y on x.


6. The scores obtained by a group of students in tests that measure verbal ability (X) and abstract reasoning (Y) are represented in the following table:

y/x 20 30 40 50
(25-35) 6 4 0 0
(35-45) 3 6 1 0
(45-55) 0 2 5 3
(55-65) 0 1 2 7

1 Is there a correlation between the two variables?

2 According to the data, if one of these students obtained a score of 70 points in abstract reasoning, what would be the estimated score in verbal ability?


7. It is determined that there is no relationship between the consumption of paper and water in a city.

1 What is the value of the covariance of these variables?

2 What is the linear correlation coefficient?

3 Determine the equations of the two regression lines and interpret the relationship.


8. The number of offenses committed in the past year by four drivers of a transport company and their respective experience in years is represented by the following table:

Years (x) 3 4 5 6
Offenses (y) 4 3 2 1

Calculate the linear correlation coefficient and interpret it.


9. A person has entered weekly football pools and has noted the number of correct predictions he has made over four weeks in February. The correct predictions are represented in the following table:

Pools (X) 6 8 6 8
No. of Correct Predictions (Y) 1 2 2 1

Determine the linear correlation coefficient and interpret it. Based on the success this individual has experienced in February, should potential betters have confidence in his predictions?