STEVENS
INSTITUTE OF TECHNOLOGY
1870
HOMEWORK 9
MA 541-B - Spring 2024
Note:
+ At the end of this assignment, you can find instructions on how to use Excel and
Minitab to run multiple linear regression. You can use any software packages to solve
the problems in this homework though.
+ Make sure to include screenshots of the software output in your work.
Problem 1:
A chain of sports clubs wishes to use regression analysis to help determine which
features should be included in their new location. They believe that median income in
the area is a significant factor in determining the number of people who join a
neighborhood sports club. The CEO of the chain gathered data from existing sports clubs
regarding the number of members each club had, the median income in the area in which
they were located, and whether the clubs had a pool, racquetball courts, or group fitness
classes. If management can determine with 90% confidence that a pool, racquetball
courts, or group fitness classes produces significantly more memberships than sports
clubs without those features, they will include them in the new location.
Sports Club Membership
Number of Median Income
Pool?
Racquetball
Fitness
Members
($)
Courts?
Classes?
1258
32223
No
No
No
1479
34975
No
No
No
1480
43187
No
Yes
No
1701
44337
No
No
No
2014
52167
No
No
Yes
2271
57521
No
No
Yes
2615
58347
No
Yes
No 2632
60960
Yes
No
No
2737
62201
Yes
No
Yes
2810
67993
No
No
Yes
3563
68770
No
No
Yes
3765
81289
Yes
Yes
Yes
3792
83902
No
No
Yes
4069
84594
Yes
No
Yes
4393
86855
Yes
Yes
Yes
4787
88381
Yes
Yes
Yes
a) What sign do you expect the correlation coefficient between the number of
members and the median income to have (without calculating)? Explain why.
b) Create three dummy variables, pool, courts, and classes, that are equal to 1 if the
observation contains this feature and equal to 0 if the observation does not contain
this feature.
c) Use statistical software to estimate the following regression models. In each case,
write the estimated regression equation and state whether the coefficient of the
independence variable is significant at the 0.10 level. (Make sure to include the
following in your answers: hypotheses Ho and Ha, test statistic value, p-value,
conclusion.)
i) Members = ẞo + ẞ1 (Pool) + εi
=
ii) Members ẞo + ẞ1 (Courts) + &i
iii) Members = ßo + ß₁ (Classes) + ɛi
d) Estimate the following multiple regression model.
Members =
ẞo + ẞ1 (Income) + ẞ2 (Pool) + ß³ (Courts) + ß4 (Classes) + &¡
Write the estimated regression equation.
e) Are any of the coefficients of the indicator variables significant at the 0.10 level?
f) Explain why it is important to include the income variable in the regression model.
g) After studying these regression results, how would you suggest the management of
the sports club chain go about building their new location? Should they use any of
the regression models you have estimated? Explain why or why not.
Problem 2:
The Supplemental Nutrition Assistance Program (SNAP) provides monthly benefits that
help eligible low-income households buy the food they need for good health. For most
households, SNAP finds account for only a portion of their food budgets, so they must
also use their own funds to buy enough food to last throughout the month. Eligible households can receive food assistance through regular SNAP or through the Louisiana
Combined Application Project (LaCAP). Using the data in the table, answer the following
questions to help predict monthly benefits to eligible households.
SNAP Benefits
Monthly
Benefit ($)
Family Size
Gross Monthly
Income
Monthly
Benefit ($)
Family Size
Gross Monthly
Income
603.41
5
3753
556.42
1
3098
560.69
3
3778
569.05
8
3707
623.24
6
3609
365.80
8
2071
416.12
5
2262
489.08
5
3166
323.90
1
1966
495.86
4
3126
418.78
4
2736
642.77
4
3933
506.46
2
3274
364.81
8
1925
552.53
2
3480
619.30
6
3736
586.46
7
3741
238.71
1
1453
637.18
8
3684
378.94
4
2538
244.49
2
1476
302.58
1
1798
507.19
5
2835
231.74
8
1189
512.56
5
2873
428.67
6
2247
312.89
4
1618
286.99
5
1460
329.05
4
1565
268.81
1
1567
243.49
6
1582
329.81
6
1622
560.37
8
3380
627.25
3
3828
599.90
3
3922
421.52
6
2782
657.09
5
3845
656.38
2
3978
394.82
5
2233
400.64
3
2493
a) Suggest a regression model that will assist SNAP administrators in providing a
monthly benefit to eligible households.
b) Fit the model that you suggested in part a. Is this model useful in predicting monthly
benefits? Justify your answer. (Make sure to include the following in your answers:
hypotheses Ho and Ha, test statistic value, p-value, conclusion.)
c) Are all independent variables in the model helpful in explaining the variation in
monthly benefits? Explain your answer.
d) Give a 95% confidence interval for average monthly benefits for a four-member
household with a gross monthly income of $2500. Interpret this interval.
e) Provide a 99% prediction interval for a four-member household with a gross monthly
income of $2500. Interpret this interval.
f) What is the difference between the intervals found in parts d and part e? Discovering Technology
Using Excel
Use the data in the table below to perform multiple regression analysis.
1. Enter the labels Delivery Time, Number of Pizzas, and Distance in cells A1, Bl, and
C1, respectively.
2. Enter the y (Delivery Time), x, (Number of Pizzas), and x, (Distance) data into columns
A, B, and C, respectively.
A
B
C
Number of
Delivery Time
Distance
Pizzas
1
2
16.68
7
5.6
3
11.5
3
2.2
4
12.03
3
3.4
5
14.88
8
0.8
6
13.75
6
1.5
18.11
7
3.3
8
8
2
1.1
9
17.83
10 79.24
11
21.5
735
2.1
30
14.6
6.05
12
40.33
16
6.88
13
21
10
14
13.5
15
19.75
946
2.15
4.00.
2.55
4.62
16
24
17
29
10
18
15.35
916
4.48
7.76
2
19
19
7
1.32
20
9.5
0.36
21
35.1
17
7.7
22
17.9
10
1.4
23
52.32
26
8.1
24
18.75
9
4.5
25
19.83
8
6.35
26
10.75
4
1.5
Figure 14.15
3. Under the Data tab, choose Data Analysis, and Regression.
4. In the dialog box, select the delivery time data in Column A for the Input Y Range
($A$1:$A$26). 5. For the Input X Range, select the data in columns B and C for number of pizzas and
distance ($B$1:$C$26). Check the box next to Labels since the column titles are
included in the selection. If you would like Excel to compute confidence intervals
for individual coefficients at a level other than 95%, you can check the box next to
Confidence level and enter the desired level of confidence as a percentage (i.e. for a 99%
confidence interval you would enter 99). Click OK.
6. Observe the summary output for the regression analysis. In the Regression Statistics
table you will find Multiple R, which is the positive square root of the coefficient of
determination, the coefficient of determination, R2, the adjusted R² value, the standard
error for the model, and the total number of observations. The ANOVA table gives the
degrees of freedom for regression and error (residual), along with SSR, SSE, TSS, MSR,
MSE, and the F-statistic. The Significance F column contains the P-value corresponding
to the F-statistic. This is the P-value we are interested in when testing the overall model
for significance. Finally, the estimated values of the coefficients are reported along
with the standard error for each coefficient, the t-statistic, P-value, and 95% confidence
interval for the individual coefficient.
A
1 SUMMARY OUTPUT
Regression Statistics
B
C
D
E
F
G
2
3
4 Multiple R
0.981846095
5 R Square
0.964021754
6 Adjusted R Square
0.960751004
7 Standard Error
3.075694294
25
8 Observations
9
10 ANOVA
901002
11
df
12 Regression
2
13 Residual
22
14 Total
24
MS
F
SS
5576.424901 2788.212451 294.7403048
208.1176986 9.45989539
5784.5426
Significance F
1.30749E-16
307
15
16
17 Intercept
18 Number of Pizzas
19 Distance
1.589101026
1.567708058
P-value
t Stat
Coefficients Standard Error
1.048779122 1.709514681 0.101425497
1.792903307
0.156345691 10.16402191 8.97113E-10
0.327534513 4.786390423 8.84845E-05
Lower 95%
Upper 95%
-0.382131459 3.967938072
1.26485991 1.913342142
0.888443055 2.246973061
Figure 14.16
Using Minitab
Regression
ORIG) x bos
hodiny
Use the data in Table 14.1 to perform multiple regression analysis.
1. Enter the labels Delivery Time, Number of Pizzas, and Distance in columns C1, C2,
and C3, respectively.
2. Enter the y (Delivery Time), x, (Number of Pizzas), and x, (Distance) data into columns
C1, C2, and C3, respectively.
3. Choose Stat, Regression, and Regression.
4. Enter C1 (Delivery Time) in the Response box, and C2 and C3 (Number of Pizzas and
Distance) in the Predictors box. Press OK.
