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Exploration of the Robustness of the t-procedure
Recall that our population of coin ages for pennies in circulation was extremely skewed to the right with
EMBED Equation.DSMT36
The t-procedure is said to be robust. We would like to test this notion. How reliable are the results we obtain when the conditions for the t-procedure are violated? We will start by taking a sample of size n = 5, and using the mean of this sample, construct a 95% confidence interval for the true mean age of the population of pennies. We will repeat this 100 times so that we have 100 confidence intervals. Since we have the good fortune of actually knowing that EMBED Equation.DSMT36 , we can count the number of confidence intervals that capture the true mean age of our population of pennies and thus test how well our procedure works.
Fathom Overview and Instructions:
Open the document coin ages robust 03.ftm
To begin the investigation, you will click on the Collect More Measures button, BUT before you do that, below is a brief explanation of whats going on.
The vertical line in the diagram represents the Population Mean, EMBED Equation.DSMT36 . If a particular confidence interval "captures" the population mean of 12.26, it is plotted with a green line; if it does not, it is plotted with a red line.
Click on the Collect More Measures button
to construct 95% confidence intervals for 100 samples of size n=5. How many of the intervals generated captured EMBED Equation.DSMT36 , the true population mean? Repeat several times.
How does the definition of a 95% confidence interval
If you collect many samples of the same size from a population and construct a 95% confidence interval for each, about 95% of the intervals would cover/capture/contain the true population mean, EMBED Equation.DSMT36
correspond to what you observed (the Summary Table shows the number and proportion of intervals that captured the true population mean)?
Investigate:
You can change the sample size by dragging on the sampleSize slider
or double clicking on the value, typing a new value and hitting return. To see the effect of your change, click on the Collect More Measures button. You might want to do this several times for each investigation.
What effect did changing the sample size have on the number of intervals that captured the true population mean? Why?
What effect did changing the sample size have on the size of the intervals? Why?
In the Estimate of Sample of 1000 Pennies box, the confidence level (in blue) can be edited by clicking on it. This brings up the formula editor. Type in a new value and click OK. To see the effect of the change, click on the Collect More Measures button.
What effect did changing the confidence level have on the number of intervals that captured the true population mean? Why?
What effect did changing the confidence level have on the size of the intervals? Why?
After completing this investigation, check your understanding of confidence intervals by reviewing the information from the website
Tools for Teaching and Assessing Statistics http://www.gen.umn.edu/research/stat_tools/
What students should understand about confidence intervals:
* A confidence interval for a population mean is an interval estimate of an unknown population parameter (the mean), based on a random sample from the population
* A confidence interval for a population mean is a set of plausible values of the parameter (m) that could have generated the observed data as a likely outcome.
* A confidence interval for a population mean consists of a sample statistic ( EMBED Equation.DSMT36 ) plus or minus a measure of sampling error (which is error from random sampling), when we have approximate normality of the sampling distribution.
* The level of confidence tells the probability the method produced an interval that includes the unknown parameter.
* The probability relates to the method (data, interval), not to the parameter.
* An increase in sample size leads to a decreased interval width: large samples have narrower widths than small samples (all other things being equal).
* Higher confidence levels have wider intervals than lower confidence levels (all other things being equal).
* Narrow widths and high confidence levels are desirable, but these two things affect each other.
* If many random samples are independently sampled from a population and 95% confidence intervals constructed for each one, we would expect about 5% of the intervals to not include the population mean (the population parameter). This 95% refers to the process of taking repeated samples and constructing confidence intervals for each.
* Confidence intervals for a population mean should be based upon a t statistic when the population distribution is approximately normal (or at least not too skewed) and s is unknown.
* A confidence interval suggests what parameter values are reasonable given the data and all values in the interval are equally plausible as values of m that could have produced the observed sample mean.
* After you calculate one confidence interval, the parameter is either included or not, but you don't know.
* It is desirable to have a narrow width (for more precise estimates) with a high level of confidence. A narrow width alone is not sufficient (if it has a low level of confidence).
II. What students should be able to DO with this knowledge:
* Know how to make a confidence interval wider or narrower (what factors can be changed)
* Know how to compute a confidence interval for a mean given sample data
* Know how to interpret a confidence interval, make an appropriate inference (in context) and be able to make a correct probability statement as an interpretation of a confidence
III. Some common misconceptions students should NOT have:
* there is a 95% chance the confidence interval includes the sample mean
* there is a 95% chance the population mean will be between the two values (upper and lower limits)
* 95% of the data are included in the interval
* a wider interval means less confidence
* A narrower confidence interval is always better (regardless of confidence level)
PAGE 5
Ruth E. Carver
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