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Lab 5.3: Estimation, Prediction and Tolerance

Objectives

To use sampling from a known population to illustrate confidence, prediction, and tolerance intervals.

Lab Procedure

I.
The Population

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.

A.
Open SASDATA.STATOPOLIS 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). You will see that there are four variables in the data set:

o
HHSIZE: household size.
o
VALUEH: the value of the house.
o
H_INCOME: household income.
o
H_GENDER: gender of the head of the household (0=female, 1=male)

B.
Do a distribution analysis on H_INCOME (by choosing Analyze: Distribution ( Y )).

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 $\alpha=2.3$ and $\beta=25000$. The density curve for this gamma distribution is

\begin{eqnarray*} p(y) & = & (6.57\times 10^{-11}) y^{1.3}e^{-y/25000},~y>0,\\ & = & 0,~y\leq 0. \end{eqnarray*}



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.

II.
Selecting Samples and Obtaining Data

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 (1) estimate the population mean household income using a confidence interval, (2) predict a new household income drawn from the population using a prediction interval, and (3) obtain a range of values that with high probability contains at least 95% of all household incomes in the population using a tolerance 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_3) for you to analyze in Part III of this lab.

A.
Select the samples by running the SAS macro LAB5_3.2 The samples of size 5, 50 and 2 will be written to the SAS data sets WORK.SAMP5, WORK.SAMP50, and WORK.NEWOBS, respectively.

B.
Open each of the samples in SAS/INSIGHT. For the samples of size 5 and 50,

1.
Compute the mean, $\overline{y}$, and a 95% confidence interval for the population mean $\mu=57500$. To do this, choose Analyze: Distribution( Y ) and input H_INCOME as the the Y variable. From the resulting analysis window, select Tables: Basic Confidence Intervals: 95%. The first row of the 95% Confidence Intervals table contains $\overline{y}$ (under Estimate) and the confidence interval endpoints (LCL and UCL). Now evaluate whether this interval contains the true population mean $\mu=57500$. Write down these four quantities for both the SAMP5 and SAMP50 data sets.

2.
For each sample, compute a level 0.95 prediction interval for a new observation. To do this, obtain the sample mean, $\overline{y}$, the sample variance, $s^2$, and the square of the standard error of the mean, $s^2/n$. The first two are found in the Moments table in the SAS: Distribution window. The second is obtained by squaring the quantity labelled Std Mean in that table. Use the $s^2$ and $s^2/n$ values to compute the estimated standard error of prediction using the formula


\begin{displaymath}\hat{\sigma}(Y_{new}-\hat{Y}_{new})=\sqrt{s^2+s^2/n}\end{displaymath}

Now compute the prediction interval using the formula


\begin{displaymath}\overline{y}\pm \hat{\sigma}(Y_{new}-\hat{Y}_{new}) t_{n-1,0.975}\end{displaymath}

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.

3.
For the samples of size 5 and 50, compute a normal theory level 0.95 tolerance interval for a proportion 0.99 of the population values. Use formula (5.27), p. 269 of the text, or you can use the SAS macro NORTOL.

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 $\alpha=2.3$ and $\beta=25000$, and interval endpoints $A=5000$ and $B=190000$. 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.

4.
When you have completed parts 1-3 above for each sample, 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_3.

III.
Analysis

Open the SAS data set SASDATA.LAB5_3 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:

o
LCL5: The lower confidence limit from the sample of size 5.

o
UCL5: The upper confidence limit from the sample of size 5.

o
INCL5: 1 if the confidence interval from the sample of size 5 includes the population mean; 0 otherwise.

o
LCL50: The lower confidence limit from the sample of size 50.

o
UCL50: The upper confidence limit from the sample of size 50.

o
INCL50: 1 if the confidence interval from the sample of size 50 includes the population mean; 0 otherwise.

o
LPL5: The lower prediction limit from the sample of size 5.

o
UPL5: The upper prediction limit from the sample of size 5.

o
NEW5: The new observation corresponding to the sample giving the prediction interval from the sample of size 5.

o
INPL5: 1 if the prediction interval from the sample of size 5 includes the corresponding new observation; 0 otherwise.

o
LPL50: The lower prediction limit from the sample of size 50.

o
UPL50: The upper prediction limit from the sample of size 50.

o
NEW50: The new observation corresponding to the sample giving the prediction interval from the sample of size 50.

o
INPL50: 1 if the prediction interval from the sample of size 50 includes the corresponding new observation; 0 otherwise.

o
LTOL5: The lower tolerance limit from the sample of size 5.

o
UTOL5: The upper tolerance limit from the sample of size 5.

o
PROP5: The proportion of population values covered by the tolerance interval from the sample of size 5.

o
INTOL5: 1 if the tolerance interval from the sample of size 5 includes at least 99% of the population values; 0 otherwise.

o
LTOL50: The lower tolerance limit from the sample of size 50.

o
UTOL50: The upper tolerance limit from the sample of size 50.

o
PROP50: The proportion of population values covered by the tolerance interval from the sample of size 50.

o
INTOL50: 1 if the tolerance interval from the sample of size 50 includes at least 99% of the population values; 0 otherwise.

Have a look at these to familiarize yourself with them.

A.
Run the SAS Macro LAB5_3CI. This will produce two plots of the confidence intervals in the SASDATA.LAB5_3 data set: one for sample size 5 and the other for sample size 50. The plots are color-coded: green indicates the population mean, $\mu$ is contained in the interval, and red indicates it is not. The macro also computes the mean width of the confidence intervals. Print the plots and write down the values of the mean widths for submission with your lab report.

Two issues in the performance of confidence intervals are coverage and precision.

1.
Coverage refers to the proportion of intervals that contain the true parameter value. Calculate the coverage from the confidence interval plots for sample sizes 5 and 50 for submission with your lab report. Are they both close to the nominal coverage of 0.95? To each other?

2.
Precision refers to interval width. Compare the mean interval widths for both sample sizes. Theory says that the width should be proportional to $1/\sqrt{n}$ (since the standard error of the mean is $\sigma/\sqrt{n}$). Is this the case here? Justify your answer.

B.
Run the SAS Macro LAB5_3PI. This will produce two plots of the prediction intervals in the SASDATA.LAB5_3 data set: one for sample size 5 and the other for sample size 50. The plots are color-coded: green indicates the appropriate new observation (NEW5 or NEW50) is contained in the interval, and red indicates it is not. The macro also computes the mean width of the prediction intervals. Print the plots and write down the values of the mean widths for submission with your lab report.

The two issues of coverage and precision are also important for the analysis of prediction intervals.

1.
For prediction intervals, coverage refers to the proportion of intervals that contain their corresponding new observation. Calculate the coverage from the confidence interval plots for sample sizes 5 and 50 for submission with your lab report. Are they both close to the nominal coverage of 0.95? To each other?

2.
As it does for confidence intervals, precision of prediction intervals refers to interval width. Compare the mean interval widths for both sample sizes. Theory says that the width should be proportional to $\sqrt{1+1/n}$ (since the standard error of the prediction error is $\sigma\sqrt{1+1/n}$). Is this the case here? Justify your answer.

C.
Run the SAS Macro L5_3TI5. This will produce a plot of the tolerance intervals in the SASDATA.LAB5_3 data set based on the samples of size 5. The plot has two parts: one showing the intervals and the second showing the proportion of population values contained within each interval. Both parts are color coded: green indicates that at least 99% of the population values lie between the endpoints of the interval, and red indicates the percentage is less than 99. The macro also computes the mean width of the tolerance intervals. Print the plot and write down the values of the mean widths for submission with your lab report.

D.
Run the SAS Macro L5_3TI50. This macro produces the same output as L5_3TI5, but for samples of size 50. Print the plot and write down the values of the mean widths for submission with your lab report.

The two issues of coverage and precision are also important for the analysis of tolerance intervals.

1.
For prediction intervals, coverage means that the interval contains at least the desired proportion of population values (here the proportion is 0.99). Calculate the coverage from the confidence interval plots for sample sizes 5 and 50 for submission with your lab report. Are they both close to the nominal coverage of 0.95? To each other?

2.
As it does for the other types of intervals, precision of tolerance intervals refers to interval width. Compare the mean interval widths for both sample sizes.

E.
Based on what you have seen in parts III. A.-D., summarize how sample size affects coverage and precision for the three types of intervals.

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. Which of the intervals you evaluated do you think might have been affected by the nonnormality of the population distribution? In what way were they affected? Explain your choices.

IV.
Lab Report Checklist

In your lab report, be sure to include the following:

$\Box$
Histogram of population values with density curve (I.A.).

$\Box$
For the confidence intervals you compute by hand: (1) The sample size (5 or 50) (2) The sample mean, $\overline{y}$ (3) The interval (4) Whether it contains the population mean, $\mu$. (II.B.1.).

$\Box$
For the prediction intervals you compute by hand: (1) The sample size (5 or 50) (2) The interval (3) The new value to be predicted (4) Whether it contains the new value. (II.B.2.).

$\Box$
For the tolerance intervals you compute by hand: (1) The sample size (5 or 50) (2) The interval (3) The proportion of population values it contains (4) Whether it contains at least 99% of all population values. (II.B.3.).

$\Box$
Two plots, mean widths for confidence intervals, and coverage (III.A.).

$\Box$
Two plots, mean widths for prediction intervals, and coverage (III.B.).

$\Box$
Two plots, mean widths for tolerance intervals, and coverage (III.C. and D.).

$\Box$
Overall summary of findings (III.E.).

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Joseph D Petruccelli 2001-12-04