Stat 1000 Tips about Confidence Intervals for the Mean
Published: Sun, 11/14/10
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Tips about Confidence Intervals for the Mean
These tips apply to HW9 for those of you doing the distance/online version of the course. I assume the rest of you will also have similar questions on your next assignment.
Study Lesson 8 in my book, if you have it, to prepare for this topic.
First, be sure to note whether a question gives you σ, the population standard deviation, or s, the sample standard deviation. That dictates whether you will use z or t when making your confidence interval. I would assume, at this stage, you are likely to be given σ most of the time.
Question 1 is using the formula I introduce in question 6 of Lesson 8. Make sure you look through my questions 6, 7 and 8 before attempting this question. Also, take a look at my question 10 for an example of how to deal with an unusual level of confidence.
Question 2 is standard stuff. Note, the margin of error in a confidence interval is everything after the +/-.
Question 3 requires you go to http://bcs.whfreeman.com/ips6e/
and click the "Statistical Applets" button and then click "Confidence Interval". To graph your results, use JMP 8 (do not use Crunchit!, it can't draw the graph requested). Open a "New Data Table" and name your first column something like "Confidence Intervals". In that column you will type 10, 20, 30, ... 200. Double-click at the top to the right of your first column to create a second column and name it "Percent Hit" and record the percent hits for your intervals as given by the applet. You are expecting the Percent Hit to eventually be 95% by the end. Graph your data as a time series. Here's how:
To make a Time Series: Select Analyze in the toolbar, then select Modeling in the drop-down list and finally select time series. Select your time variable ("Confidence Intervals") and click "X, Time ID" and select the other variable you are tracking ("Percent Hit") and click "Y, Time Series". Click OK. Just ignore that other pop-up menu asking about time lags or autocorrelations or whatever, click OK and move on. None of that has anything to do with the time series.
Personally, I think they should have had you click the "Sample 50" button instead and made you do that several times (say up to a total of 1000 or 2000 confidence intervals). Try that yourself. Notice, in the long run, you reach a 95 percent hit rate. That is because, we know a 95% confidence interval for the mean μ will catch μ 95% of the time. Which is to say, if we make 2000 confidence intervals for μ, then we would expect about 95% of them to actually contain μ.
Question 4 requires you to copy and paste the data into JMP 8 (or Crunchit!). Personally, I would use JMP since you had to use JMP in question 3 anyway. Again, click "New Data Table", then select "Edit" then "Paste with Column Names". Now select "Analyze", "Distribution" and highlight "DRP" and click "Y, Columns", then click OK. You are now looking at a histogram and stuff. Click the red triangle next to "DRP" then select "Confidence Interval" from the drop-down list. Select "Other" to get a pop-up menu. It probably already has 0.95 typed in (for 95% confidence interval), but, if not, be sure to type in 0.95. Make sure you click the box saying "Use known Sigma". Click "OK" and you will then get a pop-up menu to type in the sigma value. I believe σ = 11 in your case. Click "OK" and JMP gives you the Confidence Interval at the bottom of the printout.
If you want to use Crunchit! Copy the "http:..." stuff for the link to the data (shown at the top in your web browser) then select "Data" and "Load Data from URL". Paste the link into the box and make sure the box saying "use first line as column names" is selected. Make sure you also click the drop-down menu next to "Delimiter" and select "whitespace". Now, you should see the data given in two columns in your spreadsheet. Select "Statistics", then select "Z tests", then select "One-sample". In the pop-up window, make sure you click "drp" in the "Variable" box to highlight it. Type the given standard deviation into the box (again, σ = 11) and make sure the slider is set at 95% confidence and click "OK". You now get a printout showing, among other things, the 95% CI.
In either case (JMP or Crunchit!), you can now select, copy and paste your output to a file ready for upload as usual.