A research team from the Department of Economics at UC-Davis wants to investigat

Question

A research team from the Department of Economics at UC-Davis wants to investigate whether sales in a community (Y) can be predicted from the number of teenagers in a community (X1) and the per capita disposable personal income in the community (X2) and collected the following data from 21 counties in CA: X1 (# in thousands), X2 (in $1000) and Y (in $1000). Before any modeling, try preliminary data analysis (e.g., descriptive statistics or data check). Fit regression model(s). Explain the Fitted model, esp., interpretation of regression parameters. Cheek or diagnose model fits. Assume you have the following new data (which are not from original data set): Compute the predicted value and the CI's for E[Y(X1, X2)] and Y(X1, X2), where Y(a, b) denotes a function of a and b. What is the expected or predicted sales when the number of teenagers are 60,000 and per capita disposable income is $50,000? Comment your result. What do you feel about your analysis? Any comments? If you are the team leader, what issues do you want discuss with team members and if/how you will do differently (e.g., design and/or analysis)?

Explanation / Answer

The raw data given is as below:

166.5

a. Here is the descriptive statistics for the given data:

b) Here is the fitted regression model for the given data:

INTERPRET REGRESSION STATISTICS TABLE

The below table gives the overall goodness-of-fit measures:
      R2 = 0.916
      Correlation between y and y-hat is 0.9574 (when squared gives 0.916)

Adjusted R2 = R2 - (1-R2 )*(k-1)/(n-k)=0.907 where n=no of observations

The standard error here refers to the estimated standard deviation of the error term u.
It is sometimes called the standard error of the regression. It equals sqrt(SSE/(n-k)).

The fitted line from the above table is given as :

Y=-68.86+1.45*X1+ 9.36* X2 ----@

c) To diagnose the model fit , we will check the below residual output. Here residuals are the difference between actual and predicted value which lies in the range of (-18.4,20.21)

d) Based on the equation calculated above (eq @), the predicted value of Y is:

The expected value of Y can be just calculated by multiplying the probability with the above predicted value and CIs:

e) the expected and predicted sales when no of teenagers are 60000 (X1=60) and income is 50000 dollars(X2=50)

Calculated from the equation generated above.

f) As a team leader, the analysis can be refined more. The avilable sample is very small to fit a model and then generalize. Needed large sample data to use more sophisticated techniques in the data. Over-fitting could have been avoided. The prediction can be made stronger by finding out hidden factors apart from available variables.

x1 x2 y 68.5 16.7 174.4 45.2 16.8 164.4 91.3 18.2 244.2 47.8 16.3 154.6 46.9 17.3 181.6 66.1 18.2 207.5 49.5 15.9 152.8 52 17.2 163.2 48.9 16.6 145.4 38.4 16 137.2 87.9 18.3 241.9 72.8 17.1 191.1 88.4 17.4 232 42.9 15.8 145.3 52.5 17.8 161.1 85.7 18.4 209.7 41.3 16.5 146.4 51.7 16.3 144 89.6 18.1 232.6 82.7 19.1 224.1 52.3 16

166.5

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