Consumer Research, Inc., is an independent agency that conducts research on consumer attitudes and behaviours for a variety of firms. In one study, a client asked for an investigation of consumer...

Task 1 10 Marks
Consumer Research, Inc., is an independent agency that conducts research on consumer attitudes and behaviours for a variety of firms. In one study, a client asked for an investigation of consumer characteristics that can be used to predict the amount charged by credit card users. Data were collected on annual income, household size, and annual credit card charges for a sample of 50 consumers. The following data are recorded for Consumer information.
    Income ($1000s)
    Household Size
    Amount Charged ($)
    Income ($1000s)
    Household Size
    Amount Charged ($)
    54
    3
    4016
    54
    6
    5573
    30
    2
    3159
    30
    1
    2583
    32
    4
    5100
    48
    2
    3866
    50
    5
    4742
    34
    5
    3586
    31
    2
    1864
    67
    4
    5037
    55
    2
    4070
    50
    2
    3605
    37
    1
    2731
    67
    5
    5345
    40
    2
    3348
    55
    6
    5370
    66
    4
    4764
    52
    2
    3890
    51
    3
    4110
    62
    3
    4705
    25
    3
    4208
    64
    2
    4157
    48
    4
    4219
    22
    3
    3579
    27
    1
    2477
    29
    4
    3890
    33
    2
    2514
    39
    2
    2972
    65
    3
    4214
    35
    1
    3121
    63
    4
    4965
    39
    4
    4183
    42
    6
    4412
    54
    3
    3720
    21
    2
    2448
    23
    6
    4127
    44
    1
    2995
    27
    2
    2921
    37
    5
    4171
    26
    7
    4603
    62
    6
    5678
    61
    2
    4273
    21
    3
    3623
    30
    2
    3067
    55
    7
    5301
    22
    4
    3074
    42
    2
    3020
    46
    5
    4820
    41
    7
    4828
    66
    4
    5149
Required:
1. Use methods of descriptive statistics to summarize the data. Comment on the findings.
2. Develop estimated regression equations, first using annual income as the in- dependent variable and then using household size as the independent variable. Which variable is the better predictor of annual credit card charges? Discuss your findings.
3. Develop an estimated regression equation with annual income and household size as the independent variables. Discuss your findings.
4. What is the predicted annual credit card charge for a three-person household with an annual income of $40,000?
5. Discuss the need for other independent variables that could be added to the model. What additional variables might be helpful?
Solution:
1.
I will first analyse the three variables separately, to see how the data is distributed.
    Income ($1000s)
    
    Household Size
    
    Amount Charged ($)
    
    
    
    
    
    
    
    
    Mean
    43.48
    
    Mean
    3.42
    
    Mean
    3963.86
    Standard Error
    2.057786
    
    Standard Error
    0.24593
    
    Standard Error
    132.0234
    Median
    42
    
    Median
    3
    
    Median
    4090
    Mode
    54
    
    Mode
    2
    
    Mode
    3890
    Standard Deviation
    14.55074
    
    Standard Deviation
    1.738989
    
    Standard Deviation
    933.5463
    Sample Variance
    211.7241
    
    Sample Variance
    3.024082
    
    Sample Variance
    871508.7
    Kurtosis
    -1.24772
    
    Kurtosis
    -0.72281
    
    Kurtosis
    -0.74248
    Skewness
    0.095856
    
    Skewness
    0.527896
    
    Skewness
    -0.12886
    Range
    46
    
    Range
    6
    
    Range
    3814
    Minimum
    21
    
    Minimum
    1
    
    Minimum
    1864
    Maximum
    67
    
    Maximum
    7
    
    Maximum
    5678
    Sum
    2174
    
    Sum
    171
    
    Sum
    198193
    Count
    50
    
    Count
    50
    
    Count
    50
Income_($1000s)
65.0
60.0
55.0
50.0
45.0
40.0
35.0
30.0
25.0
20.0
Income_($1000s)
Frequency
8
6
4
2
0
Std. Dev = 14.55
Mean = 43.5
N = 50.00
Household_Size
7.0
6.0
5.0
4.0
3.0
2.0
1.0
Household_Size
Frequency
16
14
12
10
8
6
4
2
0
Std. Dev = 1.74
Mean = 3.4
N = 50.00
Amount_Charged ($)
5750.0
5500.0
5250.0
5000.0
4750.0
4500.0
4250.0
4000.0
3750.0
3500.0
3250.0
3000.0
2750.0
2500.0
2250.0
2000.0
1750.0
Amount_Charged ($)
Frequency
10
8
6
4
2
0
Std. Dev = 933.49
Mean = 3964.1
N = 50.00
Since the standard deviation is a high percentage of the mean, all the three data sets are significantly spread. The income data appears to be uniformly distributed according to the histogram. The household size data is skewed, due to the fact that the two persons compose majority of households. The amount charged data appears to be normally distributed.
With the help of the correlation matrix, I will analyse the sample correlation coefficients between each pair of variables:
    
    
    Income ($1000s)
    Household size
    Amount Charged
    Income ($1000s)
    
    
    
    
    Pearson Correlation
    1.000
    .173
    .631
    
    
    
    
    
    
    Sig. (2-tailed)
    .
    .231
    .000
    
    Household size
    
    
    
    
    Pearson Correlation
    .173
    1.000
    .753
    
    
    
    
    
    
    Sig. (2-tailed)
    .231
    .
    .000
    
    Amountcharged
    
    
    
    
    Pearson Correlation
    .631
    .753
    1.000
    
    
    
    
    
    
    Sig. (2-tailed)
    .000
    .000
    .
    
Income_($1000s)
Household_Size
Amount_Charged ($)
There is correlation between amount charged and income and between amount charged and household size, also it is possible to conclude with very high confidence level (above 99%) that there is no relationship between income and household size, as the scatter matrix plot suggests.
2. Simple linear regression: Amount Charged vs. Annual Income
Y = 2204 + 40.48X
Where Y = Amount charged in $
X = Income in thousands of $
Significance of model:
    
    Sum of Squares
    df
    Mean Square
    F
    Sig.
    Regression
    
    
    
    
    
    16999744
    1
    16999744
    31.751
    .000
    
Model is significant with more than 99% confidence level.
Residual analysis also shows no particular pattern and no problems of autocorrelation.
    R
    R Square
    Adjusted R Square
    Std. Error of the Estimate
    .631
    
    
    
    .398
    .386
    732
    
About 40% of the variation in Amount Charged is explained by annual income.
The standard error of the estimate is significant part of the possible predicted values within the range; it is about...