# Sampling Confidence Intervals Worksheet

March 14, 2016

Question
Summer II 2016: STA 210-221
Sampling Confidence Intervals Worksheet
Derek Gutierrez
July 18, 2016
The reasons why a confidence interval is constructed the way it is because (1) the statistic is our best guess at
estimating the true parameter and so the interval is centered on the statistic and (2) the MOE quantifies the
error in the statistic and thus the MOE is place on each side of the statistic.
This worksheet is a variation of Beyond the Numbers 2.19 in the workbook and is meant to demonstrate what
is means to be K% confident. Recall that confidence tells us how many confidence intervals, out of a long list,
are expected to contain the true parameter. That is, for a K% CI referring to a sample of size n, approximately
K% of all samples of size n will produce confidence intervals that contain the true parameter.
Instructions: Suppose we have some fair dice and we want to know out of an infinite number of rolls of the
dice, what percentage of the time we will roll a 1, 2, 3, or 4. We take a sample of size n = 80 and calculate the
percentage of times we see a 1, 2, 3, or 4 out of the 80 rolls. We then have the following:
Population: All
possible
rolls
of
the
dice
Sample: 80
rolls
of
the
dice
Parameter: The true percentage of all rolls that the dice would show a 1, 2, 3, or 4
Statistic: The percentage of 1, 2, 3, or 4 faces that we see out of a sample of 80
NOTE: This is a special case where the true parameter is known. Given the dice are fair the parameter is 2/3.
We will be simulating 95% confidence intervals. In theory, if we construct a long list of 95% confidence
intervals then about 95% of those intervals will contain the parameter.
1. Go to the following website: http://www.rossmanchance.com/applets/ConfSim.html
2. Under “Methods” make sure you have “Proportions”, “Binomial”, and “Score/Adjusted Wald” as the
settings.
3. Set to be 0.67.
4. Set n to be 80. This is the simulated sample size.
5. Set “Intervals” to be 100. This will allow us to select 100 samples of size 80 and compute a confidence
interval for each.
6. Set “Conf level” to be 95%.
7. Click “Sample”.
Note that the green lines represent intervals that cover the parameter. The red lines represent intervals that do
not cover the parameter.
1. About what percentage of the confidence intervals cover the true parameter?

2. In theory, what percentage of the confidence intervals would be expected to cover the parameter based
on 95% confidence?

3. Clearly articulate how “confident” you can be that a 95% confidence interval will contain the parameter?

4. Repeat the steps on page 1 of the worksheet except this time change “Conf level” to 82%. About what
percentage of the confidence intervals cover the true parameter?

5. In theory, what percentage of the confidence intervals would be expected to cover the parameter based
on 82% confidence?

6. Clearly articulate how “confident” you can be that a 82% confidence interval will contain the parameter?

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