Today’s lab will concentrate on understanding more about confidence intervals, their construction and interpretation. The first part is a group activity with individual responsibility that is to be completed in the lab period. The second part involves computer simulations to explore what confidence intervals mean and what effect changing the confidence level has on the confidence interval.
1. The population we will be working with consists of all females in STAT 101 who filled out a questionnaire at the beginning of the spring 1999 semester. The response variable is the height (in cm) of the female. The 250 heights are given on the accompanying page. The standard deviation for this population of heights is σ = 7 .
6 cm. The process for constructing a confidence interval for the mean height of all STAT 101 females is:
• select a random sample of size n
• calculate the sample mean X
• construct the confidence interval
X
− z ∗
σ
√ n
, X + z ∗
σ
√ n where z ∗ corresponds to the chosen level of confidence.
Exercises:
(a) Use the table of random numbers in your text book to select a random sample of n = 9 heights.
Each height in the table is identified by a row and a column. If you get the random number 125, you should go to row 12 and column 5 to obtain the height (175 cm).
Note: You can make more efficient use of the 3 digit random numbers by using the following scheme. For numbers between 000 and 249 go directly to the table. For numbers between 250 and
499 subtract 250 and then go to the table. For numbers between 500 and 749 subtract 500 and then go to the table. For numbers between 750 and 999 subtract 750 and then go to the table.
(b) Calculate the sample mean for the 9 heights in your sample. Is it the same as the sample means for the other members in your group? Why?
(c) Using your sample mean and σ = 7 .
6 cm, construct an 80% confidence interval for the mean height of all females in our population. Write your confidence interval on the blackboard.
(d) Graph your confidence interval on your answer sheet. Graph the other confidence intervals listed on the black board on your answer sheet.
i. Are all the lines representing confidence intervals the same length? Explain why, or why not.
ii. Why are the centers of the confidence intervals at different positions?
(e) One cannot say which of the confidence intervals capture the population mean and which miss it unless one knows the value of the population mean. Once you have plotted all the intervals, ask the lab instructor for the value of the population mean. Draw a horizontal line on your plot corresponding to this value.
i. How many of the intervals cover, or capture, the population mean?
ii. How does this compare to the confidence level of 80%?
(f) Given what you have seen, explain the meaning of the phrase: “80% confident”.

Your name
(a) Random sample of 9 heights random number row column height (cm)
(b) Sample mean height.
(c) 80% confidence interval for the population mean height.

(d) i. Are all the lines representing confidence intervals the same length? Explain why, or why not.
ii. Why are the centers of the confidence intervals at different positions?
(e) i. How many of the intervals cover, or capture, the population mean?
ii. How does this compare to the confidence level of 80%?
(f) Given what you have seen, explain the meaning of the phrase: “80% confident”.

The second part of the lab involves computer simulations to explore what confidence intervals mean and what effect changing the confidence level has on the confidence interval. You may work alone or in groups but each individual should hand in an answer sheet.
Go to a computer that has access to the World Wide Web. Start Netscape (if you have an old version of Netscape or use a different Web browser you may encounter difficulties, especially if your browser does not support frames) and go to the following address: http://www.public.iastate.edu/
∼ wrstephe/stat101L.html
Under Computing , click on Confidence Intervals . The top portion of this page explains a little bit about the simulation. In the middle of the page is the actual simulation. There is a number line ranging from .01 to .05. Below the number line is a slide that allows you to select a value of alpha.
Alpha is equivalent to 1 minus the confidence level. So for a 98% confidence interval, you should choose an alpha of 1-.98=.02. The graphic will display confidence intervals for 50 randomly selected samples, each vertical line represents the confidence interval for a single sample. The solid horizontal line represents the population mean, which is set at zero. Below, a counter keeps track of how many confidence intervals do not cover the population mean.
Exercises:
(a) At what confidence level does the simulation start?
(b) Are all of the lines representing confidence intervals the same length? Explain why, or why not.
(c) Why are the centers of the confidence intervals at different positions?
(d) How many of the 50 intervals do not cover the mean? How many do cover the mean?
(e) Besides the counter at the bottom, how do you know an interval does not cover the mean?

(f) Set the slide for a 98% confidence interval, and ask for the number of samples in the table below by pressing the more intervals button repeatedly. For each, fill in the number of intervals that do not contain the mean and calculate the percent that do contain the mean.
not
(g) Repeat part (f) for a 95% confidence interval. Click on new alpha to reset the simulation.
not
(h) Now, move the slider back and forth a few times. What do you notice about the intervals?
(i) Given what you have seen today, explain the meaning or the phrase “95% confident.”