The SAS data set SASDATA.STATOPOLIS contains information on 100,000 households.1For the purposes of this lab, these 100,000 households will constitute the population.
Notice that the density histogram has many bars. H_INCOME takes so many different values, it is easier to model its distribution using a density curve. To see what such a curve might look like, select Curves: Kernel Density then click OK. Print or save this histogram with the density curve for your lab report.
By using some statistical trickery, we have managed to come up with a standard density curve that models the population closely. It's called a gamma distribution with parameters and . The density curve for this gamma distribution is
The gamma distribution is common in probability and statistics, and probabilities involving it may be computed using the SAS macro NPROBS, which you will do in Part II of this lab. In the rest of the lab, we will assume this gamma distribution is the population distribution.
In this part of the lab, you will take two random samples from the population: one of size 5, and one of size 50. You will use the data in the size 5 and size 50 samples to obtain a range of values that with high probability contains at least 99% of all household incomes in the population using a tolerance interval. After computing these quantities on the data you sampled, you will pool your results with those of others in the class. This pooled data will be used in this lab next term to evaluate the performance of the three kinds of intervals Since this is a new lab, we have created a pooled data set (under the name SASDATA.LAB5_3TI) for you to analyze in Part III of this lab.
Once you have obtained the tolerance interval, check whether it really contains at least 99% of all population household incomes. To do this, use the SAS macro NPROBS. The following illustrates how:
Suppose the tolerance interval you obtained has endpoints 5,000 and 190,000. Access the macros by selecting Solutions: EIS/OLAP Application Builder: Applications: Run Private Applications. Select the macro NPROBS. In the macro window, choose the gamma distribution with parameters and , and interval endpoints and . The resulting value is 0.9849, meaning that 98.49% of all household incomes lie between $5000 and $190000. therefore, this tolerance interval fails to contain at least 99% of all household incomes in the population.3
For each of the data sets your group generates, write down the tolerance interval, the proportion of the population values it contains, and whether it contains at least 99% of all household incomes in the population. Submit the results to the TA. The values for the entire class will be input to a SAS data set for use next term. Because this is a new lab, we have created a data set of 100 observations for you. You will find it in the SAS data set SASDATA.LAB5_3TI.
Open the SAS data set SASDATA.LAB5_3TI in SAS/INSIGHT now (Recall that to get into SAS/INSIGHT you choose Solutions: Analysis: Interactive Data Analysis from any of the main SAS windows). The data set has the following variables:
Have a look at these to familiarize yourself with them.
Two issues in the performance of tolerance intervals are coverage and precision.
The population distribution of H_INCOME is nonnormal. In fact, it's pretty heavily right skewed. For some types of intervals this will make a large difference and for some it will make little difference. Do you think the performance of the tolerance intervals might have been affected by the nonnormality of the population distribution? In what way were they affected? Explain your choices.
In your lab report, be sure to include the following:
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