Construct and Interpret Confidence Intervals for Means Using Simulations

When you calculate a point estimate in statistics, there is always uncertainty (variability) around that estimate because this value is based on a sample taken from the population of interest. A confidence interval provides a range of possible values for a parameter based on a data set. The level of confidence tells us the probability that the interval will contain the actual parameter value. Confidence intervals are used in a variety of fields such as clinical trials, market research and business to name a few.

In this practical activity, you will investigate the properties of a confidence interval for a mean using simulations. Using the StatCrunch applet, you will understand the impact of the confidence level, how the shape of population distribution may impact the properties of the confidence interval, and how increasing the sample size may impact the properties of the confidence interval.

Instructions

I will send log in info to go into satcrunch and follow the directions once you get started

Part I:

CONDUCT THIS ACTIVITY IN MYLAB BY PEARSON

1. Select “Content” from the Brightspace Navigation menu.
2. From the “Content” menu, select the “MyLab Statistics” tab.
3. Click on the “MTH 210 StatCrunch” tab
4. Click on “Open Link”
5. Go to the second arrow: Click on Visit the StatCrunch Website
6. Click on the Orange > Open StatCrunch
7. Click on Applets > Confidence Intervals > For a Mean
8. When the Applet loads, 100 confidence intervals for a population mean appear in the plot in a stacked fashion. Within the plot, the value for the true mean displays as a vertical black line. Green intervals contain this mean, but red intervals don’t.
9. You can hit “Reset” to generate more intervals (ie. 1000 intervals), change sample size and change the confidence level. The associated confidence intervals for a mean are appended to the result. The table above the graph shows the cumulative proportion of the confidence intervals that contain the mean.

NORMAL  DISTRIBUTION: GENERATE 1000 CONFIDENCE INTERVALS CHANGING THE CONFIDENCE LEVEL AND THE SAMPLE SIZE

1. Select the normal distribution and hit “Compute!” (Keep:  mean=50, standard deviation 10, 0.95 confidence, Sample size: 10 and T interval for the first simulation).  Once it has created the intervals, hit Reset.

1. Now select 1000 intervals. Click on Options > copy > rt click to copy image and save your intervals and table in a Word document. Now click on “Analyze” and from the pull-down menu Stat> Summary Stats> column and select Lower bounds and under statistics select “Min” and Compute! Do this again but this time, select the column Upper bounds and under statistics, select “Max” and Compute! Copy and paste both tables into a Word document.

1. Reset and repeat again for 0.99 confidence keeping everything else the same. Record the same information as listed above in #2.
2. Hit Reset.  Now change the confidence level back to 0.95 and type in Sample size: 50.  Record all the information as you did in #2 above.
   1. Hit Reset.  Change the confidence to 0.99 keeping the Sample size: 50.  Record the same information you did in #2 above.

SKEWED DISTRIBUTION: GENERATE 1000 CONFIDENCE INTERVALS CHANGING THE CONFIDENCE LEVEL AND THE SAMPLE SIZE

1. Close out your intervals and summary statistics from the Normal Distribution and go back to the Instructions (step 4) above.
2. Select the Right Skewed Distribution and hit Compute! (Keep everything as is: mean=50, standard deviation 10, 0.95 confidence, Sample size: 10 and T interval for the first simulation).  Once it has created the intervals. Hit Reset.
   1. Now select 1000 intervals. Click on Options > copy > rt click to copy image and save your intervals and table in a Word document. Find your maximum upper and minimum lower bounds for your Confidence Intervals by doing the following: Click on “Analyze” and from the pull-down menu select Stat> Summary Stats> column and select Lower bounds and under statistics select “Min” and Compute! Do this again but this time, select the column Upper bounds and under statistics, select “Max” and Compute! Copy and paste both tables into a Word document.
   1. Reset and repeat again for 0.99 confidence keeping everything else the same. Record the same information as listed above in #3.
   1. Hit Reset.  Now change the confidence level back to 0.95 and type in Sample size: 50.  Record all the information as you did in #3 above.
   1. Hit Reset.  Change the confidence to 0.99 keeping the Sample size: 50.  Record the same information you did in #3 above.

Part II: Analysis

ANSWER THE FOLLOWING QUESTIONS

Normal Distribution

1. What is the proportion of the 95% confidence intervals that contain 50 when n=10? When n=50?

1. What is the proportion of the 99% confidence intervals that contain 50 when n=10? When n=50?
2. How does the typical width of the confidence interval change when the confidence level increases?
   1. How does the typical width of the confidence interval change as the sample size increases?

Right Skewed Distribution

1. What is the proportion of the 95% confidence intervals that contain 50 when n=10? When n=50?

1. What is the proportion of the 99% confidence intervals that contain 50 when n=10? When n=50?
2. How does the typical width of the confidence interval change when the confidence level increases?
   1. How does the typical width of the confidence interval change as the sample size increases?
   1. What was one obvious difference when comparing your conference intervals when sampling from a Normal Distribution vs. a Skewed Distribution? What was consistent (similar) when increasing the sample size for both Normal and Skewed Distributions?