Fill in the blanks:

8.1.      The magnitude of the correlation is indicated by the correlation _____ which can range from -1.00 to +1.00.

8.2.      The most common and efficient way to present the correlations of several variables with each other is by using a(n) ______ table.

8.3.      The correlation between two variables can be shown graphically by a ________.

8.4.          The null hypothesis predicts that the correlation coefficient is equal to _______.

8.5.           The Spearman rank order correlation is used when the variables to be correlated are measured on a(n)  ______ scale.

Circle the correct answer:

8.6.      The hypothesis that states that r¹0 is an example of a(n) alternative/null hypothesis.

8.7.      When an increase in one variable is associated with a decrease in the other variable, the correlation between these two variables is positive/negative.

8.8.      In order to use the Pearson product-moment correlation, the variables to be correlated should be measured on an ordinal/interval scale.

8.9.      When the points on a scattergram go from the bottom left to the top right they represent a positive/negative correlation.

8.10.       The true correlation between two variables may be underestimated when the variance of one of the variables is very high/very low.

8.11.       When the null hypothesis is rejected at p<.001, it means that the chance that r=0 is very small/very high.

8.12.       The null hypothesis is rejected when the obtained correlation coefficient is higher/lower than the critical value.

Answer/compute the following questions:

8.13     Which correlation coefficient (a or b) shows a stronger relationship between the two variables being correlated?

a.         X1&Y1: r =  .85

b.         X2&Y2: r = -.94

8.14.    Following are two scattergrams (in Figure A and in Figure B). Four different correlation coefficients are listed under each scattergram. Choose the coefficient that best matches each scattergram.

Y                                                                       Y  ·                                                                            ·  ·

·            ·                                                                              ·  ·

·                 ·                                                                ·    ·

·             ·                                                                  ·    ·

·                ·                                                          ·  ·    ·

·                                                                                ·   ·   ·

·               ·                                                       ·  ·

·                ·                                                             ·    ·

·                ·                                                    ·    ·

·                                                           ·    · X X

Figure A                                                         Figure B

A1. r= .50                                                      B1. r= -.57

A2. r= .78                                                      B2. r=   .92

A3. r= -.10                                                      B3. r=   .38

A4. r= -.89                                                      B4. r=  -.91

8.15     Following is a scattergram showing the scores of 8 statistics students on two tests, X and Y. Each of the first 7 students is represented by a dot and their scores are listed in the table that follows. Use the scattergram to find the scores of student #8 on test X and test Y. The location of this student on the scattergram is represented by a large dot (•) next to number 8.

Y 1                             4 4                           ·                              ·

5          2 3                           ·          ·

6          8 2                           ·          ·

7          3 1               ·          ·      X 1          2        3        4        5

Student #       X         Y

1              2          4

2              3          3

3              2          1

4              5          4

5              2          3

6              2          2

7              1          1

8              ?          ?

8.16     What do these two scattergrams have in common?

Y                                                              Y  ·   ·    ·  ··    ·   ··   ·    ·

·   ·    ·  ··    ·   ··   ·     ·

·   ·    ·  ··    ·   ··   ·    ·  ··                                    ·   ·    ·  ·  ·  ·  ··  ····  ·

·   ·    ·  ··    ·   ··   ·    ·  ··

·   ·    ·  ··    ·   ··   ·    ·

·   ·    ·  ··    ·   ··   ·    ·  ··

·    ·  ··    ·   ··   ·    ·  ··  8.17     Estimate (do not calculate!) the correlation between the advertising spending and sales that were obtained over a 5 year span. Indicate whether the correlation is positive or negative, and whether it is high or low. Explain your answer.

Year                        Ad Spending              Sales

1                               \$21,000                   \$83

2                                 15,000                      70

3                                 17,000                      68

4                                 25,000                      90

5                                 19,000                      74

8.18     Estimate (do not calculate!) which of the two sets of consumer research studies (A&B or X&Y) has a higher correlation. Explain your answer.

Set 1                                                               Set 2

Study #           A         B                                 Study #           X         Y

1                      41        50                                1                      66        47

2                      41        47                                2                      53        36

3                      38        43                                3                      50        45

4                      30        39                                4                      48        38

5                      28        37                                5                      45        39

8.19     Match the correlation coefficient and the diagram illustrating this correlation. a.     r = .91

b.     r = .28

c.     r = .15

8.20.    Study the following intercorrelation table. Two surveys measure consumer attitudes and two surveys measure marketers’ promotional spending. Knowing that Test 1 measures consumer attitudes, speculate which is the other consumer attitudes survey (2, 3, or 4) and which two surveys measure marketers’ promotional spending. (A hint: the two consumer attitudes surveys should correlate higher with each other than with the two marketers’ promotional spending surveys, and the two marketers’ promotional spending surveys should correlate higher with each other than with the two consumer attitudes surveys.)

 2 3 4 1 .35 .89 .23 2 .39 .92 3 .34

8.21.        Following are results from a study correlating gross domestic product (GDP) and the Consumer Price Index (CPI) from a group of 16 Asian countries, a group of 15 European countries, and the two groups combined.

 Group Correlation (r)/ Group Size (n)/ Significance (p value) Asia r =.59 n = 16 p =.02 Europe r =.52 n = 15 p =.05 Combined r =.53 n = 31  p =.01

a.        Which correlation coefficient is the highest?

b.        Which correlation coefficient has the highest statistical significance?

c.        How can a correlation of r=.53 (from the combined group) be more statistically significant (better p value) than a correlation of r=.59 (from the Asian countries)?

Chapter 9

PREDICTION AND REGRESSION

Fill in the blanks:

9.1.      In simple regression, there is/are _____ predictor(s).

9.2.      The regression line is also called a line __________.

9.3.      The slope of the regression line is represented by the letter ___.

9.4.      When the regression equation is used to draw a line, the point where that line intersects the vertical line (the Y-axis) is represented by the letter ___ which indicates the intercept.

9.5.      When the correlation between two variables is perfect and positive, and we use one of these variables to predict the other one, the standard error of estimate (SE) is _____.

9.6.      The difference between an actual Y score and its corresponding predicted Y score (Y’) is called the _____ score.

9.7.      In multiple regression with two predictors, there is/are _____  intercept(s), represented in the equation by the letter a.

Circle the correct answer:

9.8.      In regression, the predictor is called the independent/dependent variable, and the predicted variable (or the criterion variable) is called the independent/dependent.

9.9.      The predicted variable is represented by the letter X/Y and the predictor is represented by the letter X/Y.

9.10.    In the regression equation, the letter b represents the constant/coefficient and the letter a represents the constant/coefficient.

9.11.    The dependent variable can be predicted more accurately as the correlation between the independent and dependent variables increases/decreases.

9.12.    As the correlation between the predictor and the criterion variable increases, the standard error of estimate (SE) increases/decreases.

9.13.    The predicted Y scores are expected to be on/around the regression line.

9.14.    The criterion is predicted more accurately when SE is larger/smaller.

9.15.    Study the following graph. If a company is able to maintain a 25% gross margin (the predictor X), what is the company’s predicted Return on Investment (ROI) (the criterion Y)?

9.16.    Compute SE (the standard error of estimate) when the standard deviation of the Y-variable is 5 (SDY=5) and the correlation is 0.00 (r=0.00). Use the following equation to compute the SE:

What is the relationship between the SE (the standard error of estimate) and the SD of the dependent variable Y (SDY=5) when the correlation is zero (r=.00)? Explain.

9.17.    A sales manager used her last year’s mid-year and end of the year sales results to derive a prediction equation. This year her sales force has the same 6 salespeople as last year and the market conditions are very similar to last year.  In the middle of this year the manager wants to predict the year-end sales results for each of her 6 salespeople. Following are the mid-year sales results of the 6 salespeople, and the prediction equation.

a.          Calculate the salespeople’s’ predicted end of the year sales results (Y’ scores). (Note: Use the prediction equation to compute the students’ Y’ scores.)

b = 1.5             a = 13.2

9.18.Study the following graph, and determine what is the value of the slope of the regression line (i.e., b). Explain your answer.

9.19.Draw a regression line in the following graph, when the intercept is 10 (a=10), the mean of the X-variable is 20 (=20), and the mean of the Y-variable is 30 (=30).

9.20.    Figure A shows three predictors, X1, X2, and X3, and their correlations with the criterion variable Y1. Figure B shows three predictors, Z1, Z2, and Z3, and their correlations with a criterion variable Y2.

a.         Which predictor variables, those depicted in Figure A or those depicted in Figures B, correlate higher with each other?

b.         Which set of three predictors, those shown in Figure A (X1, X2, and X3) or those shown in Figure B (Z1, Z2, and Z3) is likely to predict the criterion variables (Y1 or Y2) more accurately?

Figure A                                                          Figure B