QA 233 (Basic Business Statistics)

Solutions to Practice Problem Set X

Dee Zerta is the proprietor of Ice Dreams, a local chain of ice cream soda shops. Ms. Zerta frequently has difficulty estimating her shop’s daily demand, which in turn has lead to problems with her inventory. Because vanilla ice cream is the basis for so many of Ice Dreams’ menu items (such as banana splits, parfaits, sundaes, shakes and malts), anticipation of demand for this flavor is her greatest challenge. However, Ms. Zerta has noticed that her sales seem to increase as the outside temperature increases, and she believes she may be able to use this knowledge to alleviate this problem. She randomly selects twenty days from the previous year and uses the internet to find the official high temperature in her area for those days. She then uses her business records to find the amount of vanilla ice cream sold by the Ice Dreams shops on those days. Her results are given below.

Please use these data to help Ms. Zerta determine the relationship between outside temperature and sales of vanilla ice cream at her shops.

First we recognize that we wish to be able to predict or estimate Gallons of Vanilla Ice Cream Sold, so this is the dependent variable y. Since we wish to use information about Official High Temperature in making such predictions or estimations, this is our independent variable x. We will need to perform some summary calculations:

 

Date	Gallons of Vanilla Ice Cream Sold (y)	Official High Temperature (x)	xy	x2	y2
11-01-00	151	64°	9664	4096	22801
03-11-00	98	58°	5684	3364	9604
05-19-00	233	82°	19106	6724	54289
01-30-00	61	48°	2928	2304	3721
12-14-99	58	54°	3132	2916	3364
04-14-00	201	75°	15075	5625	40401
06-22-00	311	93°	28923	8649	96721
06-07-00	322	89°	28658	7921	103684
02-08-00	164	61°	10004	3721	26896
11-19-99	111	57°	6327	3249	12321
10-22-00	192	78°	14976	6084	36864
05-22-00	275	90°	24750	8100	75625
09-09-00	345	99°	34155	9801	119025
03-28-00	221	71°	15691	5041	48841
06-25-00	391	103°	40273	10609	152881
07-02-00	382	98°	37436	9604	145924
10-29-00	168	77°	12936	5929	28224
08-19-00	363	101°	36663	10201	131769
08-02-00	370	97°	35890	9409	136900
04-22-00	215	80°	17200	6400	46225
Totals	4632	1575	399471	129747	1296080

Now we can use these summary calculations to compute stimated slope coefficient b₁ and the estimated y-intercept b₀:

 

and

so the estimated regression equation is:

So we estimate that i) when Official High Temperature increases by one unit (degree Fahrenheit)  the Gallons of Vanilla Ice Cream Sold increases by 6.07 units (gallons) and ii) when Official High Temperature is zero units (degrees Fahrenheit)  the Gallons of Vanilla Ice Cream Sold is expected to be -246.501 units (this meaningless result is due to fact that we had to extrapolate to estimate the y-intercept term b₀).

-- QA 233 students - stop here for this problem! --

Although we were not asked to do so, we could also evaluate the correlation coefficient (r) and the coefficient of determination (r²) for this problem. We would first need to compute ss_yy:

Now we can calculate the correlation coefficient (r):

The correlation coefficient is close to 1.0 and positive, so we have evidence that a strong positive relationship exists between Gallons of Vanilla Ice Cream Sold and Official High Temperature. We can also calculate the coefficient of determination (r²):

Thus the regression has been able to explain 94.3% of all variation in the observed values of the dependent variable Gallons of Vanilla Ice Cream Sold in our sample data.

2. George Steinbrenner, the owner of the New York Yankees, is arguing with other about whether a baseball team’s attendance is related to how successful the fans expect the team to be. In an attempt to make his point, Mr. Steinbrenner has one of his staff members assemble data on the Yankees total attendance during each season from 1976 through 1990. He also has the staff member record the number of wins the Yankees had achieved during the previous season for each of these seasons. The data are:

   Season	Attendance (in millions of fans)	Wins During Previous Season
   1976	2.0	83
   1977	2.1	97
   1978	2.3	100
   1979	2.5	100
   1980	2.6	89
   1981	2.4	103
   1982	***	***
   1983	2.3	79
   1984	1.8	91
   1985	2.2	87
   1986	2.3	97
   1987	2.4	90
   1988	2.6	89
   1989	2.2	85
   1990	2.0	74

   Mr. Steinbrenner’s assistant recognized that approximately one-third of the 1982 season was cancelled due to a player’s strike, and so has omitted that season from the data. Please use the data collected to determine if Mr. Steinbrenner has a supportable conjecture with regards to the Yankees. Also explain for Mr. Steinbrenner the nature of the relationship between the Yankees’ attendance and their level of success on the field during the previous season.

   First we recognize that we wish to be able to predict or estimate Attendance (in millions of fans), so this is the dependent variable y. Since we wish to use information about Wins During Previous Season in making such predictions or estimations, this is our independent variable x. We will need to perform some summary calculations:

    

Now we can use these summary calculations to compute he estimated slope coefficient b₁ and the estimated y-intercept b₀:

and

so the estimated regression equation is:

So we estimate that i) when Wins During Previous Season increases by one unit (game won)  the Attendance (in millions of fans) increases by 0.0093 units (millions of fans) or 93000 fans and ii) when Wins During Previous Season is zero units (no wins)  the Attendance (in millions of fans) is expected to be 1.424 units or 1,424,000 fans (this meaningless result is again due to fact that we had to extrapolate to estimate the y-intercept term b₀).

-- QA 233 students - stop here for this problem! --

Although we were not asked to do so, we could also evaluate the correlation coefficient (r) and the coefficient of determination (r²) for this problem. We would first need to compute ss_yy:

Now we can calculate the correlation coefficient (r):

The correlation coefficient is not close to 1.0 and positive, so we have evidence that a somewhat weak positive relationship exists between Attendance (in millions of fans) and Wins During Previous Season. We can also calculate the coefficient of determination (r²):

Thus the regression has been able to explain 11.3% of all variation in the observed values of the dependent variable Attendance (in millions of fans) in our sample data.

3. Johnny Bravo, ace QA 233 student, wants to demonstrate the relationship between the number of hours a student spends studying and the grades that he/she earns. He randomly selects ten students and asks them to record the number of hours they spend studying throughout the quarter. After the quarter has concluded he also records the GPA each student earned during the quarter. His results follow:

Student ID Number	Total Hours Spent Studying During the Quarter	Grade Point Average
2196	153	3.75
2907	122	3.15
3108	97	2.40
2562	182	4.00
4680	115	2.33
1741	75	1.80
0850	104	2.50
9123	119	2.90
3408	98	2.50
2597	122	2.85

 Please use the data collected by Johnny to determine the nature of the relationship between the number of hours a student spends studying and the grades that he/she earns.

First we recognize that we wish to be able to predict or estimate Grade Point Average, so this is the dependent variable y. Since we wish to use information about Total Hours Spent Studying During the Quarter in making such predictions or estimations, this is our independent variable x. We will need to perform some summary calculations:

 

Student ID Number	Total Hours Spent Studying During the Quarter (x)	Grade Point Average (y)	xy	x2	y2
2196	153	3.75	573.75	23409	14.0625
2907	122	3.15	384.30	14884	9.9225
3108	97	2.40	232.80	9409	5.7600
2562	182	4.00	728.00	33124	16.0000
4680	115	2.33	267.95	13225	5.4289
1741	75	1.80	135.00	5625	3.2400
0850	104	2.50	260.00	10816	6.2500
9123	119	2.90	345.10	14161	8.4100
3408	98	2.50	245.00	9604	6.2500
2597	122	2.85	347.70	14884	8.1225
Totals	1187	28.18	3519.60	149141	83.4464

Now we can use these summary calculations to compute estimated slope coefficient b₁ and the estimated y-intercept b₀:

and

so the estimated regression equation is:

So we estimate that i) when Total Hours Spent Studying During the Quarter increases by one unit (hour studied)  the student's Grade Point Average increases by 0.0212 and ii) when Total Hours Spent Studying During the Quarter is zero units (no studying)  the student's Grade Point Average is expected to be 0.304 (this result is again due to fact that we had to extrapolate to estimate the y-intercept term b₀).

-- QA 233 students - stop here for this problem! --

Although we were not asked to do so, we could also evaluate the correlation coefficient (r) and the coefficient of determination (r²) for this problem. We would first need to compute ss_yy:

Now we can calculate the correlation coefficient (r):

The correlation coefficient is close to 1.0 and positive, so we have evidence that a strong positive relationship exists between Total Hours Spent Studying During the Quarter and the student's Grade Point Average. We can also calculate the coefficient of determination (r²):

Thus the regression has been able to explain 91.7% of all variation in the observed values of the dependent variable student's Grade Point Average in our sample data.

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