# Math 201: Computation of Correlation Coefficient

The data below are the final exam scores of 10 randomly selected statistics students and the number of hours they studied for the exam. Find the correlation coefficient between hours studied and final exam scores. Is there a weak or strong, positive or negative correlation between hours studied and final exam scores?

Hours x 3 5 2 8 2 4 4 5 6 3

Scores y 65 80 60 88 66 78 85 90 90 71

(Show work)

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#### Solution Preview

Pearson r correlation coefficient can be used in the data.

N(sumXY) - (sumX) (sumY)

Pearson r formula = ---------------------------------------------------

Sqrt (N(sumX^2) - (sumX)^2) (N(sumY^2) - (sumY)^2)

x x2 y y2 xy

3 9 65 4225 195

5 25 80 6400 400

2 4 60 3600 120

8 64 88 7744 704

2 4 66 4356 132

4 16 78 6084 312

4 16 85 7225 340

5 25 90 8100 450

63 3969 71 5041 4473

sum 96 4132 ...

#### Solution Summary

The solution computes for the correlation coefficient between final exam scores of 10 randomly selected statistics students and the number of hours studied. Based on the given data, it was found out that there is negligible relationship between hours studied and final exam scores.

Statistics Problems 14.7, 14.25, 14.41, 14.43, 14.49

See attached files.

14.7 The business problem facing the director of broadcasting operations for a television station was the issue of standby hours (i.e. hours in which unionized graphic artists are paid but are not actually involved in any activity) and what factors were related to standby hours. The study included the following variable:

Standby hours (Y)-Total number of standby hours in a week

Total staff present (X1)-Weekly total of people-days

Remote hours (X2)-Total number of hours worked by employees at locations away from the central plant

Data were collected for 26 weeks; these data are organized and stored in Standby.

a. State the multiple regression equation.

b. Interpret the meaning of the slopes b1 and b2, in this problem.

c. Explain why the regression coefficient, b0, has no practical meaning in the context of this problem.

d. Predict the standby hours for a week in which the total staff present have 310 people-days and the remote hours are 400.

e. Construct a 95% confidence interval estimate for the mean standby hours for weeks in which the total staff present have 310 people-days and the remote hours are 400.

f. Construct a 95% prediction interval for the standby hours for a single week in which the total staff present have 310 people-days and the remote hours are 400.

14.25 In Problem 14.3 on page 532, you predicted the durability of a brand of running shoe, based on the forefoot shock-absorbing capability (FOREIMP) and the change of impact properties over time (MIDSOLE) for a sample of 15 pairs of shoes. Use the following results:

Variable Coeffieient Stand Error t Stat p-value

INTERCEPT -0.02686 0.06905 -0.39 0.7034

FOREIMP 0.79116 0.06295 12.57 0.0000

MIDSOLE 0.60484 0.07174 8.43 0.0000

a. Construct a 95% confidence interval estimate of the population slope between durability and forefoot shock-absorbing capability.

b. At the 0.05 level of significance, determine whether each independent variable makes a significant contribution to the regression model.On the basis of these results, indicate the independent variables to include in this model.

14.41 The marketing manager of a large supermarket chain faced the business problem of determining the effect on the sales of pet food of shelf space and whether the product was placed at the front (=1) or back (=0) of the aisle. Data are collected from a random sample of stores. The results are shown inthe following table (and organized and stores in Petfood):

Store Shelf space(ft) Location Weekly Sales (dollars)

1 5 Back 160

2 5 Front 220

3 5 Back 140

4 10 Back 190

5 10 Back 240

6 10 Front 260

7 15 Back 230

8 15 Back 270

9 15 Front 280

10 20 Back 260

11 20 Back 290

12 20 Front 310

For (a) through (m), do not include interaction term.

a. State the multiple regression equation that predicts sales based on shelf space and location.

b. Interpret the regression coefficients in (a).

c. Predict the weekly sales of pet food for a store with 8 feet of shelf space situated at the back of the aisle. Construct a 95% confidence interval estimate and a 95% prediction interval.

d. Perform a residual analysis on the results and determne whether the regression assumptions are valid.

e. Is there a significant relationship between sales and the two independent variables (shelf space and aisle position) at the 0.05 level of significance?

f. At the 0.05 level of significance, determine whether each independent variable makes a contribution to the regression model. Indicate the most appropriate regression model for this set of data.

g. Construct and interpret 95% confidence interval estimates of th population slope for the relationship between sales and shelf space and between sales and aisle location.

h. Compate the slope in (b) with the slope for the simple linear regression model of Problem 13.4 on page 481. Explain the difference in results.

i. Compute and interpret the eaning of the coefficient of multiple determination, r squared.

j. Compute and intrepret the adjusted r cubed.

k. Compare r squared with the r squared vaiue computed in Problem 13.14 on page 487.

l. Compute the coefficients of partial determination and interpret their meaning.

m. What assumption about the sope of shelf space with sales do you need to make in this problem?

n. Add an interaction term to the model and, at the 0.05 level of significance, determine whether it makes a significant contribution to the model.

o. On the basis of the results of (f) and (n), which model is most appropriate? Explain.

14.43 The owner of a moving company typicall has his most experienced manager predict the total number of labor hours that will be required to complete an upcoming move. This approach has proved useful in the past, but the owner has the business objective of developing a more accurate method of predicting labor hours. In a preliminary effort to provide a more accurate method, the owner has decided to use the number of cubic feet moved and whether there is an elevator in the apartment building as the independent variables and has collected data for 36 moves in which the orign and destination were within the borough of Manhattan in NYC and the travel time was an insignificant portion of hours worked. The data are organized and stored in Moving. For (a) through (k), do not include an interaction term.

a. State the multiple regression equation for predicting labor hours, using the number of cubic feet moved and whether there is an elevator.

b. Interpret the regression coefficients in (a).

c. Predict the labor hours for moving 500 cubic feet in an apartment building that has an elevator and construct a 95% confidence interval estimate and a 95% prediction interval.

d. Perform a residual analysis on the results and determine whether the regression assumptions are valid.

e. Is there a significant relationship between labor hours and the two independent variables (cubic feet moved and whether there is an elevator in the apartment building) at the 0.05 level of significance.

f. At the 0.05 level of significance, determine whether each independent variable makes a contribution to the regression model. Indicate the most appropriate regression model for this set of data.

g. Construct a 95% confidence interval estimate of the population slope for the relationship between labor hours and cubic feet moved.

h. Construct a 95% confidence interval estimate for the relationship between labor hours and the presence of an elevator.

i. Compute and interpret the adjusted r squared.

j. Compute the coefficients of partial determination and interpret it's meaning.

k. What assumption do you need to make about the slope of labor hours with cubic feet moved?

l. Add an interaction term to the model and, at the 0.05 level of significance, determine whether it makes a significant controbution to the model.

m. On the basis of results of (f) and (l), which model is more appropriate? Explain.

14.49 The director ofa training program for a large insurance company has the business objective of determining which training method is best for training underwriters. The three methods to be evaluated are traditional, CD-ROM based, and Web based. The 30 trainees are divided into 3 randomly assigned groups of 10. Before the start of the training, each trainee is given a proficiency exam that measures mathematics and computer skills. At the end of the training, all students take the same end-of-training exam. The results are organized and stored in Underwriting. Develop a multiple regression model to predict the score on the end-of-training exam, based on the score on the proficiency exam and the method of training used. For (a) through (k), do not include an interaction term.

a. State the multiple regression equation.

b. Interpret the regression coefficients in (a).

c. Predict the end-of-training exam score for a student with a proficiency exam score of 100 and who had Web based training.

d. Perform a residual analysis on your results and determine whether the regression assumptions are vaild.

e. Is there significant relationship between the end-of-training exam score and the independent variables (proficiency score and training method) at the 0.05 level of significance?

f. At the 0.05 level of significance, determine whether each independent variable makes a contribution to the regression model. Indicate the most appropriate regression model for this set of data.

g. Construct and interpret 95% confidence interval estimate of the population slope for the relationship between end-of-training exam score and proficiency exam.

h. Construct and interpret 95% confidence interval estimates of the population slope for the relationship between end-of-training exam score and type of training method.

i. Compute and interpret the adjusted r squared.

j. Compute the coefficients of partial determination and interpret its meaning.

k. What assumption about the slope of proficiency score with end-of-training exam score do you need to make in this problem?

l. Add interaction terms to the model and, at the 0.05 level of significance, determine whether any interaction terms make a significant contribution to the model.

m. On the basis of the results of (f) and (l), which model is more appropriate? Explain.