To use sampling from a known population to
illustrate confidence intervals.
- 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
- 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:
- HHSIZE: household size.
- VALUEH: the value of the house.
- H_INCOME: household income.
- H_GENDER: gender of the head of the household (0=female, 1=male)
- 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 and
. The density curve for this gamma
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
- Selecting Samples and Obtaining Data
In this part of the lab, you will take two random samples
from the population: one of size 5, and one of size 50,
which you will use to estimate the population mean household
income using a confidence 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 confidence intervals you
calculated. Since this is a new lab, we have created a
pooled data set (under the name SASDATA.LAB5_3CI) for you
to analyze in Part III of this lab.
- Select the samples by running the SAS macro
LAB5_3.2 The samples
of size 5 and 50 will be written to the SAS data sets
WORK.SAMP5 and WORK.SAMP50, respectively.
- Open each of the samples in SAS/INSIGHT. For the
samples of size 5 and 50, compute the mean,
, and a 95% confidence interval for the
population mean . 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
(under Estimate) and the confidence
interval endpoints (LCL and UCL). Now evaluate whether
this interval contains the true population mean
. Write down these four quantities for both the
SAMP5 and SAMP50 data sets, 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_3CI.
Open the SAS data set SASDATA.LAB5_3CI 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
- LCL5: The lower confidence limit from the
sample of size 5.
- UCL5: The upper confidence limit from the
sample of size 5.
- INCL5: 1 if the confidence interval from the
sample of size 5 includes the population mean; 0
- LCL50: The lower confidence limit from the
sample of size 50.
- UCL50: The upper confidence limit from the
sample of size 50.
- INCL50: 1 if the confidence interval from the
sample of size 50 includes the population mean; 0
Have a look at these to familiarize yourself with them.
- 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,
, 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.
- 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?
- Precision refers to interval width.
Compare the mean interval widths for both
sample sizes. Theory says that the width
should be proportional to
(since the standard error of the mean is
). Is this the case here?
Justify your answer.
- Based on what you have seen in part III.
A., summarize how sample size affects coverage
and precision of confidence intervals.
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 confidence
intervals. Do you think the skewness affected
the coverage of the confidence intervals you
- Lab Report Checklist
In your lab report, be sure to include the following:
- Histogram of population values with
density curve (I.B.).
- For the confidence intervals you
compute by hand: (1) The sample size (5 or 50) (2)
The sample mean, (3) The interval
(4) Whether it contains the population mean,
- Two plots, mean widths for confidence
intervals and comparison with theoretical, and
- Overall summary of findings (III.B.).
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