Simple Random Sampling

Suppose that, in a junior high school, there are a total of 4,000 students in grades 7, 8, and 9. You want to know how household income and the number of children in a household affect students’ average weekly spending for ice cream.

In order to answer this question, you draw a sample by using simple random sampling from the student population in the junior high school. You randomly select 40 students and ask them their average weekly expenditure for ice cream, their household income, and the number of children in their household. The answers from the 40 students are saved as the following SAS data set IceCream:

```data IceCream;
input Grade Spending Income Kids @@;
datalines;
7   7  39  2   7   7  38  1   8  12  47  1
9  10  47  4   7   1  34  4   7  10  43  2
7   3  44  4   8  20  60  3   8  19  57  4
7   2  35  2   7   2  36  1   9  15  51  1
8  16  53  1   7   6  37  4   7   6  41  2
7   6  39  2   9  15  50  4   8  17  57  3
8  14  46  2   9   8  41  2   9   8  41  1
9   7  47  3   7   3  39  3   7  12  50  2
7   4  43  4   9  14  46  3   8  18  58  4
9   9  44  3   7   2  37  1   7   1  37  2
7   4  44  2   7  11  42  2   9   8  41  2
8  10  42  2   8  13  46  1   7   2  40  3
9   6  45  1   9  11  45  4   7   2  36  1
7   9  46  1
;
```

In the data set IceCream, the variable Grade indicates a student’s grade. The variable Spending contains the dollar amount of each student’s average weekly spending for ice cream. The variable Income specifies the household income, in thousands of dollars. The variable Kids indicates how many children are in a student’s family.

The following PROC SURVEYREG  statements request a regression analysis:

```title1 'Ice Cream Spending Analysis';
title2 'Simple Random Sample Design';
proc surveyreg data=IceCream total=4000;
class Kids;
model Spending = Income Kids / solution;
run;
```

The PROC SURVEYREG  statement invokes the procedure. The TOTAL=4000 option specifies the total in the population from which the sample is drawn. The CLASS statement requests that the procedure use the variable Kids as a classification variable in the analysis. The MODEL statement describes the linear model that you want to fit, with Spending as the dependent variable and Income and Kids as the independent variables. The SOLUTION option in the MODEL statement requests that the procedure output the regression coefficient estimates.

Figure 90.1 displays the summary of the data, the summary of the fit, and the levels of the classification variable Kids. The "Fit Statistics" table displays the denominator degrees of freedom, which are used in F tests and t tests in the regression analysis.

Figure 90.1 Summary of Data
 Ice Cream Spending Analysis Simple Random Sample Design

The SURVEYREG Procedure

Regression Analysis for Dependent Variable Spending

Data Summary
Number of Observations 40
Mean of Spending 8.75000
Sum of Spending 350.00000

Fit Statistics
R-square 0.8132
Root MSE 2.4506
Denominator DF 39

Class Level Information
Class Variable Levels Values
Kids 4 1 2 3 4

Figure 90.2 displays the tests for model effects. The effect Income is significant in the linear regression model, while the effect Kids is not significant at the 5% level.

Figure 90.2 Testing Effects in the Regression
Tests of Model Effects
Effect Num DF F Value Pr > F
Model 4 119.15 <.0001
Intercept 1 153.32 <.0001
Income 1 324.45 <.0001
Kids 3 0.92 0.4385

 Note: The denominator degrees of freedom for the F tests is 39.

The regression coefficient estimates and their standard errors and associated t tests are displayed in Figure 90.3.

Figure 90.3 Regression Coefficients
Estimated Regression Coefficients
Parameter Estimate Standard Error t Value Pr > |t|
Intercept -26.084677 2.46720403 -10.57 <.0001
Income 0.775330 0.04304415 18.01 <.0001
Kids 1 0.897655 1.12352876 0.80 0.4292
Kids 2 1.494032 1.24705263 1.20 0.2381
Kids 3 -0.513181 1.33454891 -0.38 0.7027
Kids 4 0.000000 0.00000000 . .

 Note: The denominator degrees of freedom for the t tests is 39.Matrix X'X is singular and a generalized inverse was used to solve the normal equations. Estimates are not unique.