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 three random samples from the population: one of size 5, one of size 50, and one of size 2. You will use the data in the size 5 and size 50 samples to predict a new household income drawn from the population using a prediction interval. You will use the sample of size 2 to check whether the prediction intervals you computed for sample sizes 5 and 50 contain a new observation from the population.
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_3pi) for you to analyze in Part III of this lab.
Now compute the prediction interval using the formula
After you obtain the first prediction interval, check whether it contains the first observation in the data set WORK.NEWOBS. After you obtain the second prediction interval, check whether it contains the second observation in the data set WORK.NEWOBS. For both the SAMP5 and SAMP50 data sets, write down the prediction interval, the corresponding new observation from WORK.NEWOBS, and whether the prediction interval contains that new observation, and 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_3PI.
Open the SAS data set SASDATA.LAB5_3PI 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 prediction intervals are coverage and precision.
The population distribution of H_INCOME is nonnormal. In fact, it's pretty heavily right skewed. Sometimes this can have an adverse effect on the coverage of prediction intervals. Do you think the skewness affected the coverage of the prediction intervals you evaluated? Explain.
In your lab report, be sure to include the following:
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