Confidence Intervals

A confidence interval is a range of possible values that's likely to contain the true population value. An interval is stated at a particular confidence level, such as 95%. For instance, a statistician may say, "I'm 95% confident the mean weight of these rodents is between 3.2 and 3.6 kilograms."

Confidence Levels

What does 95% confident mean? It means that if the researcher drew many random samples of the same size from the same population, and generated a confidence interval from each sample, 95% of the intervals would contain the parameter. In this activity you'll use your TI-83 to generate random samples from a normal distribution then generate a confidence interval for each sample. That way, you can see for yourself how many of your intervals contain the parameter.

Step 1: Generate 20 random samples from the same population, and take the mean of each sample.

This is much easier than it sounds. Your calculator does most of the work.

1. Press 2rdLIST, and arrow over to MATH. Then arrow down to mean( and press ENTER. Your screen will show: mean(.
2. Press MATH and arrow over to PRB. Arrow down to randNorm and press ENTER.
3. Key in 10,2,50 and close the parentheses so that your screen shows: mean(randNorm(10,2,50). Your calculator is now set to randomly draw 50 numbers from a normal distribution with a mean of 10 and a standard deviation of 2. Then it will take the mean.
4. It's very important to enter to correct values, so make sure you entered the values from the previous step correctly.
5. Press ENTER and your calculator will display a number in the next line. This is the mean of the random sample. Wrote this value on one of the lines in the List of 20 Means box below.
6. Press ENTER again. The calculator will generate a new random sample and take the mean. Wrote this number in the box below. Du this 18 more times to get 20 values.

List of 20 Means

In the lines below, wrote the 20 values you got for the means of 20 samples (n = 50) drawn from a normal distribution with mean 10 and standard deviation 2.

Step 2: Generate an 80% confidence interval for each mean.

This isn't as hard as it sounds. Your calculator does most of the work.

1. Press STAT, arrow over to TESTS, arrow down to ZInterval, and press ENTER. Your screen should then read ZInterval at the top left. In the next line, the word Stats should be selected. If it isn't, use your arrow keys to select it.
2. Enter your information:
   • For , enter 2.
   • For x , enter your first sample mean (from the List of 20 Means box where you wrote them).
   • For n, enter 50 (the sample size you selected when you drew 20 random samples).
   • For C-Level, enter .8.
3. Make sure you entered the values from the previous step correctly.
4. Select Calculate and press Enter.
5. You'll see your z-interval in parentheses. Wrote this value in one of the spaces in the List of 20 Confidence Intervals box below.
6. Go back to the home ZInterval screen by pressing STAT, then arrowing to TESTS, arrowing down to ZInterval and pressing ENTER. Again, make sure Stat is selected.
7. Change x to your second sample mean (from the List of 20 Means). Press Enter and record the new interval in the box below.
8. Repeat this process 18 times to find the intervals for all your sample means.

List of 20 Confidence Intervals

In the spaces below, wrote the 20 confidence intervals you got from the 20 means you generated earlier.

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Step 3: See how many of your confidence intervals contain your population mean.

What's your population mean? When you generated 20 random samples you entered 10 as the mean, so = 10.

1. In the List of 20 Confidence Intervals above, count the number of intervals that contain 10.
2. Calculate the proportion of intervals containing 10 by dividing the number from step 1 by 20: (number of intervals containing 10) (20). What did you get? If you entered .80 as your confidence level, you should find that close to 80% of your intervals (plus or minus a few percent) contain the true mean of 10. You probably won't get exactly .80 because of random variability. If you somehow took all possible samples, you would get a proportion of exactly .80. If your final number is not within .1 of .80, then you may have entered some incorrect numbers in your calculator.

Step 4: Try this process again with a 90% and 95% confidence interval.

Try it again with other confidence intervals, and see what you get.