Stats Summary
ProjectManager
summary_report.docxPROJECT PART A: Exploratory Data Analysis
Introduction
This is a report which will help the AJ DAVIS which is a department of store chain. This department has many credit customers. A sample of 50 credit customers was selected for analysis to know more about the credit customers. The analysis was done on five variables which include location of the household, income of the household, household size, and the number of years that the customer has lived in the current location and the customer’s current credit card balance on the store's credit card.
The pie chart on location as a variable
The graph above shows that 44% of the credit customers live in urban, 30% in suburban and 26% in rural locations.
The variable income summary table
|
Income |
|
|
|
|
|
Mean |
46.02 |
|
Standard Error |
1.963439 |
|
Median |
44.5 |
|
Mode |
30 |
|
Standard Deviation |
13.88361 |
|
Sample Variance |
192.7547 |
|
Kurtosis |
-1.09058 |
|
Skewness |
0.258555 |
|
Range |
49 |
|
Minimum |
25 |
|
Maximum |
74 |
|
Sum |
2301 |
|
Count |
50 |
The average income of the sample is 46.02 with a standard deviation of 13.88. The minimum of the income is 25 and the maximum is 74 which gives a range of 49.
The distribution table
|
size |
frequency |
Relative frequency |
Cumulative frequency |
|
1 |
8 |
0.16 |
0.16 |
|
2 |
7 |
0.14 |
0.3 |
|
3 |
6 |
0.12 |
0.42 |
|
4 |
4 |
0.08 |
0.5 |
|
5 |
4 |
0.08 |
0.58 |
|
6 |
6 |
0.12 |
0.7 |
|
7 |
7 |
0.14 |
0.84 |
|
8 |
8 |
0.16 |
1 |
|
total |
50 |
|
|
The bar chart of the variable size
The above graph shows the frequency graph of the household size. This shows that the family size of 1 and 8 the frequency is 8 and of size 4 and5 the frequency is 4
The pairing of location and income
|
Location |
Sum of Income ($1,000) |
|
Rural |
488 |
|
Suburban |
709 |
|
Urban |
1104 |
The graph above shows that the urban location has the highest income among the three locations and rural has the lowest.
The pairing of years and credit balance
The scatter plot shows that there is a positive relationship between number of years and the credit balance. As the number of years increase there is an increase in credit balance.
The pairing of location by size
The graph shows that the size of the rural household is larger as compare with the other two locations
Conclusion
We can conclude that the rural has large number of household size but small credit balance and also the income.
Project Part B: Hypothesis Testing and Confidence Intervals
a. The average (mean) annual income was greater than $45,000.
1. Stating the hypothesis
2. Level of significance is 5%
Critical value of z is 1.645
3. We use z-test statistics, because the sample is greater than 30.
4. The z computed value does not lie in the critical region, the null hypothesis is not rejected
5. It is reasonable to conclude that the average (mean) annual income was not greater than $45,000.
b. The true population proportion of customers who live in a suburban area is less than 45%.
1. Stating the hypothesis
2. Level of significance is 5%
Critical value of z is 1.645
3. We use z-test statistics, because the sample is greater than 30.
4. The z computed value lies in the critical region, the null hypothesis is rejected
It is reasonable to conclude that the true population proportion of customers who live in a suburban area is less than 45%.
c. The average (mean) number of years lived in the current home is greater than 8 years.
1. Stating the hypothesis
2. Level of significance is 5%
Critical value of z is 1.645
3. We use z-test statistics, because the sample is greater than 30.
4. The z computed value lies in the critical region, the null hypothesis is rejected
5. It is reasonable to conclude that the average (mean) number of years lived in the current home is greater than 8 years.
d. The average (mean) credit balance for rural customers is less than $3,200.
1. Stating the hypothesis
2. Level of significance is 5%
Critical value of z is 1.645
3. We use z-test statistics, because the sample is greater than 30.
4. The z computed value does not lie in the critical region, the null hypothesis is not rejected
5. It is reasonable to conclude that the average (mean) credit balance for rural customers is not less than $3,200.
We are 95% confidence that the mean of the population lies between 753.66 and 49868.34
We are 95% confidence that the mean of the population lies between 0.173 and 0.427
We are 95% confidence that the mean of the population lies between 8.34 and 10.856
We are 95% confidence that the mean of the population lies between 3895.152 and 4411.768
Project Part C: Regression and Correlation Analysis
1. Generate a scatterplot for income ($1,000) versus credit balance ($), including the graph of the best fit line. Interpret.
There is a positive relationship between income and credit balance as an increase in credit balance lead to increase in income
2. Determine the equation of the best fit line, which describes the relationship between income and credit balance.
3. Determine the coefficient of correlation. Interpret.
The correlation is the square root of 0.64084 which is 0.8005. This shows that there is a strong positive correlation between income and credit balance.
4. Determine the coefficient of determination. Interpret.
The R2=0.64084. This shows that 64.08% of variation in income (y) is explain by credit balance (x)
5. Test the utility of this regression model (use a two tail test with α =.05). Interpret your results, including the p-value.
The utility of this model can be tested by a t-test for beta-1. From the obtained output we can see the test statistic for that test is 9.25 with corresponding p-value 0.000. Since the p-value is smaller than the significance level α=0.05 so we can say that the model is significant
6. Based on your findings in 1–5, what is your opinion about using credit balance to predict income? Explain.
Base on my finding, I see that Size is a good predictor of Credit Balance because Credit Balance and Size seems to affect each other. As Size increase Credit Balance seems to increases also; they correlated. As the Size of the household grow so does the Credit Balance of those household also grew and increase.
7. Compute the 95% confidence interval for beta-1 (the population slope). Interpret this interval.
I am 95% confident that for each additional credit balance that the average income will go up between 0.009335 and 0.014518
8. Using an interval, estimate the average income for customers that have credit balance of $4,000. Interpret this interval.
The interval is given by (41.77, 46.61). I am 95% confident that the average mean income is between $41,770 and $46,610 that have a balance of $4,000.Using an interval, predict the income for a customer that has a credit balance of $4,000. Interpret this interval.
9. Using an interval, predict the income for a customer that has a credit balance of $4,000. Interpret this interval.
The interval is (27.11, 61.27). I am 95% confident that the average predicted income is between $27,110 and $61,270 that have a balance of $4,000.
10. What can we say about the income for a customer that has a credit balance of $10,000? Explain your answer.
Putting $10,000 into the regression model we have
Income =-3.516+0.011926*10,000=115.7483
So a customer with 10,000 credit balance, it is more than likely to have an income of $115,748.27 as per the fitted regression model
In an attempt to improve the model, we attempt to do a multiple regression model predicting income based on credit balance, years, and size.
11. Using MINITAB, run the multiple regression analysis using the variables credit balance, years, and size to predict income. State the equation for this multiple regression model.
|
Model Summary |
|||||||||
|
Model |
R |
R Square |
Adjusted R Square |
Std. Error of the Estimate |
Change Statistics |
||||
|
|
|
|
|
|
R Square Change |
F Change |
df1 |
df2 |
Sig. F Change |
|
1 |
.930a |
.865 |
.856 |
5.261 |
.865 |
98.405 |
3 |
46 |
.000 |
|
a. Predictors: (Constant), Credit Balance, Years, Size |
|||||||||
|
Coefficientsa |
|||||||||
|
Model |
Unstandardized Coefficients |
Standardized Coefficients |
t |
Sig. |
95.0% Confidence Interval for B |
||||
|
|
B |
Std. Error |
Beta |
|
|
Lower Bound |
Upper Bound |
||
|
1 |
(Constant) |
-13.186 |
3.608 |
|
-3.655 |
.001 |
-20.448 |
-5.923 |
|
|
|
Size |
.615 |
.418 |
.112 |
1.472 |
.148 |
-.226 |
1.456 |
|
|
|
Years |
1.210 |
.232 |
.395 |
5.209 |
.000 |
.742 |
1.677 |
|
|
|
Credit Balance |
.011 |
.001 |
.724 |
13.188 |
.000 |
.009 |
.012 |
|
|
a. Dependent Variable: Income |
Regression Analysis: Income ($1, 000 versus Credit Balance, Years, Size)
The regression equation is
|
ANOVAa |
||||||
|
Model |
Sum of Squares |
df |
Mean Square |
F |
Sig. |
|
|
1 |
Regression |
8171.685 |
3 |
2723.895 |
98.405 |
.000b |
|
|
Residual |
1273.295 |
46 |
27.680 |
|
|
|
|
Total |
9444.980 |
49 |
|
|
|
|
a. Dependent Variable: Income |
||||||
|
b. Predictors: (Constant), Credit Balance, Years, Size |
12. Perform the global test foruUtility (F-Test). Explain your conclusion.
From the output we can see the F-test statistics in this case is 98.41 with corresponding p-value 0. Thus the null hypothesis of insignificancy is rejected and we can conclude that the regression model is significant is predicting the dependent variable income
13. Perform the t-test on each independent variable. Explain your conclusions and clearly state how you should proceed. In particular, state which independent variables should we keep and which should be discarded.
The given output, t-test statistic for credit balance is 13.19 with corresponding p-value 0, for the size it is 1.47 with p-value 0.148 and for years it is 5.21 with p-value 0. Since the p-value for credit balance and years is smaller than 0.05 so they are significant is predicting income so we should keep them in the model but the p-value for size is greater than 0.05 implying size is not so significant in predicting the income thus we should remove this variable from the given model
14. Is this multiple regression model better than the linear model that we generated in parts 1–10? Explain.
We need to check at the R-sq which is the coefficient of determination for both the two models. The coefficient of determination for multiple regression model (86.5%) is greater than that of simple linear regression 64.08% so the multiple linear regression is explaining higher variance which implies that MLR model is better than simple linear regression model
Scatter plot of Income vs credit balance
2631 2047 3155 3913 2660 3531 2766 3769 4082 3806 4049 4073 2697 2914 4073 4310 4199 4253 3104 4293 4456 4340 4925 3178 4391 4947 3203 4354 4366 5003 3250 4402 4397 4595 4786 4888 5148 5011 5220 3257 5528 5283 5332 3304 5553 5484 3342 3788 5756 5861 27 25 25 26 30 29 33 30 32 34 35 40 30 33 42 32 43 43 33 47 35 54 42 36 57 44 38 54 54 46 40 60 58 61 61 62 49 68 57 45 71 57 64 45 74 65 47 53 66 69
Credit Balance
Income
Count of Location
Total
Rural Suburban Urban 13 15 22
A bar chart of the houshold size
1 2 3 4 5 6 7 8 8 7 6 4 4 6 7 8
Size
Frequency
Sum of Income ($1,000) by Location
Total Rural Suburban Urban 488 709 1104
Scatter plot of years and credit balance
2 2 1 2 5 3 10 4 4 6 8 9 9 11 10 4 10 10 13 10 5 11 5 13 11 6 15 8 10 6 15 11 10 13 13 14 8 14 8 16 15 9 9 17 19 10 18 18 10 10 2631 2047 3155 3913 2660 3531 2766 3769 4082 3806 4049 4073 2697 2914 4073 4310 4199 4253 3104 4293 4456 4340 4925 3178 4391 4947 3203 4354 4366 5003 3250 4402 4397 4595 4786 4888 5148 5011 5220 3257 5528 5283 5332 3304 5553 5484 3342 3788 5756 5861
Years
Credit Balance
Sum of Size by Location
Total Rural Suburban Urban 87 69 69
Location
Size Total
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