A bank that offers charge cards to customers studies the yearly purchase amount

Question

A bank that offers charge cards to customers studies the yearly purchase amount (in thousands of dollars) on the card as related to the age, income (in thousands of dollars), whether the and years of education of the cardholder. The other variables are self-explanatory. The original data set has information on 160 cardholders. Upon further examination of the data, you decide to remove the data for cardholder 129 because this is an older individual who has a high income from having saved early in life and having invested successfully. This cardholder travels extensively and frequently uses her/his charge card. The data file “Purchases” is posted on Blackboard.

a) Fit the model to data and give the least squares equation to predict yearly purchase amount.

b) Give practical interpretations of the estimates.

c) Is there sufficient evidence ( at = .05) to say that education is a useful predictor of yearly purchase amount.

d) Evaluate the overall utility of the model at =.01.

e) Find the 95% prediction interval for yearly purchase amount for 44 year old person with 20 years education and $70,000 income.

Purchase Age Income Education 0.92 36 36.1 14 1.30 47 41.2 10 1.22 48 43.3 10 0.45 27 29.4 14 1.12 45 40.5 11 1.16 43 41.4 15 1.71 56 46.5 12 1.62 50 45.3 15 1.01 40 36.9 12 1.09 40 41.6 12 1.57 54 43.8 12 1.02 41 38.6 7 1.15 42 39.3 14 1.24 44 39.5 8 1.18 39 37.7 13 1.47 49 44.3 14 0.58 30 31.8 12 1.15 41 36.9 8 1.06 43 40.8 10 1.25 47 41.5 15 1.19 44 40.7 15 1.28 46 41.5 12 1.40 50 45.9 12 1.39 47 43.7 14 1.17 43 38.9 11 0.78 37 36.4 13 1.06 41 38.3 10 1.29 51 46.5 15 1.29 49 42.5 10 0.96 36 37.0 11 1.48 55 46.2 11 0.63 30 31.9 13 0.94 36 33.7 14 0.87 38 35.1 12 0.80 37 34.0 11 1.26 44 40.6 14 1.20 44 39.3 12 1.42 52 45.5 14 1.07 42 39.5 10 0.62 33 31.3 10 0.82 33 34.0 16 0.95 40 36.7 14 1.44 52 44.7 14 1.38 49 43.0 12 0.73 34 33.4 11 1.18 43 39.6 12 1.32 50 41.8 11 1.62 58 47.6 13 1.79 58 49.5 11 0.93 39 38.3 12 1.10 41 39.8 11 1.19 43 40.1 13 1.29 45 40.6 11 1.35 47 42.6 14 1.42 49 42.6 12 0.70 32 33.2 9 1.65 57 47.2 11 1.18 47 41.8 10 0.71 33 31.5 15 1.40 52 45.0 10 1.15 45 40.5 13 1.32 48 42.7 12 1.44 49 42.6 12 1.25 47 39.7 12 1.07 42 40.1 10 1.19 47 41.3 10 0.59 30 32.3 12 1.31 51 44.3 15 1.28 50 43.1 11 0.85 36 33.7 11 1.36 48 43.3 12 1.06 41 36.7 13 1.18 46 42.4 12 0.12 22 26.3 11 1.36 47 42.9 12 0.87 40 36.3 8 0.46 29 28.8 15 0.48 28 30.3 14 1.03 40 35.3 9 0.82 36 35.7 12 1.01 38 37.0 13 0.74 34 33.5 14 0.59 28 30.5 15 0.78 34 34.1 14 1.38 47 42.9 12 1.08 41 39.1 12 0.95 37 35.8 15 0.69 30 32.6 12 0.81 35 35.3 15 2.05 58 60.3 19 1.03 37 44.4 12 0.75 30 37.1 16 1.45 47 53.0 14 1.93 59 61.7 12 1.41 46 52.2 17 0.91 33 38.3 12 1.04 33 38.8 17 1.33 41 48.1 14 1.31 44 51.2 13 1.85 59 60.0 19 1.06 34 41.3 18 1.50 46 51.4 12 1.67 50 54.3 14 1.22 43 49.8 15 1.90 51 57.2 13 1.21 43 48.8 11 0.93 33 40.5 16 1.23 37 44.2 15 1.26 43 47.8 15 1.00 34 40.6 11 1.30 41 45.5 15 1.02 38 43.7 11 1.47 42 46.4 13 1.25 41 46.2 16 1.83 53 57.5 11 1.26 39 47.2 16 1.01 34 40.9 12 0.59 28 34.6 10 2.10 62 64.8 15 1.73 53 57.9 13 1.63 47 50.8 15 1.82 51 55.3 16 1.46 48 51.8 14 1.75 50 54.2 14 2.34 68 70.1 14 1.50 54 56.3 12 1.59 52 55.3 10 1.00 33 41.5 16 2.15 72 70.5 16 1.26 40 46.2 13 1.58 47 52.0 15 2.38 63 64.7 17 1.47 45 50.1 12 1.66 49 52.4 19 1.07 37 46.0 17 1.04 37 45.5 14 1.78 51 53.7 14 1.30 41 47.5 16 1.38 47 52.6 12 1.45 48 52.8 13 1.78 55 57.7 12 1.21 38 44.8 17 1.40 45 51.6 15 1.78 51 55.5 17 1.79 51 56.5 14 1.73 50 55.6 12 1.24 44 50.2 16 1.72 57 60.0 15 2.15 68 70.7 13 1.57 46 50.2 16 2.03 59 64.8 11 1.70 51 55.2 16 1.31 43 49.5 15 1.42 45 48.7 16 1.51 45 49.2 15 1.22 41 48.5 18 1.76 51 55.1 13 0.80 32 39.9 13 1.54 47 51.6 15 0.91 35 41.0 16

Explanation / Answer

a) the results are as follows from where the least squares equation to predict yearly purchase amount is

Purchase = -0.8426 + 0.02500 Age + 0.02000 Income + 0.00875 Education

b) Practical interpretations of the estimates:

Corresponding to one year increase in age, there is on an average an increase of $25 in the purchases, holding other predictors fixed.

Corresponding to $1000 increase in income, there is on an average an increase of $20 in the purchases, holding other predictors fixed.

Corresponding to one year increase in education, there is on an average an increase of $8.75 in the purchases, holding other predictors fixed.

c) We observe that there is sufficient evidence ( at = .05) to say that education is a useful predictor of yearly purchase amount as coefficient is signficant with t=2.40, p=0.018<0.05.

d) From ANOVA results in regression we found that the model is highly signficant with F(3,156)=872.48, p=.0.000<0.01 and so the overall utility of the model at =.01 is strong

Regression Analysis: Purchase versus Age, Income, Education

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
Regression 3 23.0838 7.69459 872.48 0.000
Age 1 1.9094 1.90941 216.51 0.000
Income 1 1.1257 1.12569 127.64 0.000
Education 1 0.0506 0.05064 5.74 0.018
Error 156 1.3758 0.00882
Lack-of-Fit 154 1.3754 0.00893 44.66 0.022
Pure Error 2 0.0004 0.00020
Total 159 24.4596

Model Summary

S R-sq R-sq(adj) R-sq(pred)
0.0939106 94.38% 94.27% 93.98%

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant -0.8426 0.0561 -15.03 0.000
Age 0.02500 0.00170 14.71 0.000 3.94
Income 0.02000 0.00177 11.30 0.000 4.35
Education 0.00875 0.00365 2.40 0.018 1.30

Regression Equation

Purchase = -0.8426 + 0.02500 Age + 0.02000 Income + 0.00875 Education

e) From following results the 95% prediction interval for yearly purchase amount for 44 year old person with 20 years education and $70,000 income=(1.75241, 1.91234)= ($1752.41, $1912.34)

Prediction for Purchase

Regression Equation

Purchase = -0.8426 + 0.02500 Age + 0.02000 Income + 0.00875 Education

Variable Setting
Age 44
Income 70
Education 20

Fit SE Fit 95% CI 95% PI
1.83238 0.0404811 (1.75241, 1.91234) (1.63037, 2.03438)

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