Please complete each one of the following three questions.
1. Andy's Ice Cream Shop is located in a very warm part of Texas. You collect data about the number of gallons of ice cream sold and the temperature outside (in Celsius):
Day Ice Cream Sales (gallons) Ave Temp (C)
1 100 10
2 132 16
3 140 20
4 213 24
5 180 21
6 75 7
7 160 18
8 150 17
9 185 23
10 160 21
a. Which variable is dependent and which variable is independent?
b. Articulate a plausible explanation for why these two variables could be related. Hint: try common sense!
c. Which correlation coefficient (Spearman Rho or Pearson) would you use for this problem? Why?
d. Calculate a correlation coefficient.
e. Perform a simple linear regression. What is the resulting equation? What is the R squared?
f. Say it is 17 degrees Celsius. How many gallons of ice cream do you think you will sell?
Check Figures - R squared about .9, F ratio > 70, slope is around 7 and intercept is around 25 or so.
2. Andy is a cooperative student with Perkins Institute, a highly selective cooperative engineering and management college. At this school students spend alternate periods of school and work. His boss has asked him to examine the relationship between incoming students' ACT (American College Testing) scores (note: these range from 12 to 32) and their high school GPA and their ultimate college GPA (on a 4.0 scale) and average work performance evaluation (on a 4.0 scale) they earn while at Perkins. Andy's boss pleads "Help me determine who to accept! I can't stand to see folks fail in this program. It costs me too much. Success means both earning good college grades and working well in the company. How can I predict success in these two areas, given the information I have to work with - namely, HS GPA and ACT score?" Note that College GPAs of under 2.0 are failing and under 2.5 are weak. Also note that work evaluation averages below 3.0 are weak. If this problem and the data seem a bit vague - they are. Unfortunately, real world data is rarely clear and unambiguous!
Assume that the following data is representative of what Perkins experiences with hundreds of new students. In working the problem don't focus on individual students in this dataset.
Student ACT HS GPA College GPA Work Eval
1 24 3.4 2.1 3.3
2 28 3.7 3.2 3.7
3 29 3.9 4.00 3.0
4 31 3.5 3.50 3.7
5 30 3.2 3.10 3.0
6 29 4.0 3.95 3.3
7 30 3.6 3.70 2.8
8 28 3.8 3.60 3.8
9 23 3.2 2.90 3.6
10 28 2.9 3.20 3.0
11 20 3.7 1.95 2.2
12 19 3.3 2.70 4.0
a. Use SPSS to graph the relationships between the following pairs of variables: ACT and College GPA, HS GPA and College GPA, HS GPA and Work Eval and ACT and Work EVAL. In words what do the graphs reveal? Is there a pattern (perhaps positively or negatively sloped)? (Hint: I would suggest running a scatter plot)
b. Calculate a correlation coefficient matrix for the four variables. (hint: use the Bivariate Correlation procedure).
c. What conclusions would you draw from this data? What would you encourage Andy's boss to do? Remember, he knows a potential student's HS GPA and ACT. He wants to predict their success in college (College GPA) and success in work (Work Evaluation).
NOTE: This problem employs fictitious data. It is based, however, on a real experience the author had several years ago.
Check figures - you should generate a four by four matrix of correlations, some of which are insignificant, others are significant. Your challenge is to interpret this matrix and give Andy's boss some solid recommendations.
3. John Herr is an analyst for the Best Foods grocery chain. The firm operates four grocery stores. John is interested in knowing if the average dollar amount per purchase is identical for the four stores. John randomly selected six receipts from each of the four stores:
Store 1 Store 2 Store 3 Store 4
13.05 16.17 5.48 9.52
23.94 18.52 6.92 10.92
15.63 19.57 9.47 11.12
25.78 21.40 7.63 9.32
17.52 13.59 11.90 12.73
18.96 20.57 9.92 7.92
a. Use SPSS to perform an ANOVA on this data.
b. What are the null and alternative hypotheses?
c. Is there support in this data for the notion that average dollar amount per purchase is the same for all stores?
Hint: You cannot enter this data as four columns. Instead enter the data in two columns: Store number (1, 2, 3 or 4) and Sales (13.05, 5.48. etc.). Think about which column is a "dependent" variable and which is a "factor".
Source: Hanke, J.E. and Reitsch, A.G. (1994). Understanding Business Statistics, 2nd edition. Homewood, Ill.: Richard D. Irwin. p. 435
Check figure - I suspect you'll find an F ratio around 17 or so.
Step by step method for computing test statistic for a number of problems from correlation analysis, regression analysis, ANOVA are given in the answers.