Finding and interpreting a confidence interval for a population standard deviation given sample data
Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to find and interpret a confidence interval for a population standard deviation given sample data. Here's our problem statement: The values listed below are waiting times (in minutes) of customers at two different banks. At Bank A, customers enter a single waiting line that feeds three teller windows. At Bank B, customers may enter any one of three different lines that have formed at three teller windows. Answer the following questions.
OK, the first part says, “Construct a 99% confidence interval for the population standard deviation σ at Bank A. Notice we have our data listed here just below the problem statement. I'm gonna click on this little icon to the right so I can open my data in StatCrunch.
Now my data is here in StatCrunch, I’m ready to go ahead and find the confidence interval estimate for the population standard deviation. To do that, I'm going to go up here to Stat –> Variance Stats (because we don't have standard deviation, only variance) –> One Sample (because I'm only dealing with one sample) and With Data (because I'm given actual data here in my StatCrunch data table).
In the options window that appears, I'm gonna select the column where my data appears, in this case Bank A, select the radio button for confidence interval, and then make sure my confidence level matches that of the problem statement. Then I come down here and click Compute! to produce this lovely results window.
Now, when you're making confidence interval estimates in StatCrunch for proportions and for means, you can take the lower and upper limit from the ends of the table here and put them in the answer field. However, when you're asked to find a confidence interval for standard deviation and you're using StatCrunch to solve that, the only way you can do that is by looking at the variance and not the standard deviation. How then do we get the standard deviation?
Well, notice the variance is simply the standard deviation squared. So if we take the square root of these limits here in the table, we can get the values that we seek. To start, I'm going to copy the value here for the lower limit, and then I'm going to pull up a calculator so I can paste the number in with Ctrl+V on my keyboard. Oh, it didn't like that. Let's try this again.
I'm going to copy that number from StatCrunch, then in my calculator, put the number in, take the square root, and here we want to round to two decimal places, so now we have 0.33 for the lower limit. I repeat the procedure again for the upper limit. Now that I have limits in, I check my answer. Well done!
Now the second part says, “Construct a 99% confidence interval for the population standard deviation σ at Bank B. Well, I could go through the same process again with the menu options and what not, or I could just come to my results window here and click on the Options button located in the upper left corner. In the drop down menu that follows, I select Edit, and I'm right back in the options window that I had previously. And I just switch my data from Bank A to Bank B, hit Compute! and now I've got new numbers. I have to do the same thing again, take the square root of my upper and lower limits to find the values that I need. Good job!
And now the last part of the problem says, “Interpret the results found in the previous parts. Do the confidence intervals suggest a difference in the variation among waiting times? Does a single line system or the multiple line system seem to be a better arrangement?” Well, let's scroll back so we can see the confidence intervals for the two different banks. And if we go back to our problem statement, we see that the Bank A customers enter a single waiting line, but at Bank B customers enter a multi-line system.
So Bank A is the single-line system, and we see here that the range here is about a little less than one minute. For Bank B with the multi-line system, we see that the range for our confidence interval is just above three minutes, so a significantly longer range here for Bank B than for Bank A. That means there's much more variation here for Bank B which has the multi-line system than for Bank A which has a single-line system.
So the single-line system will have lower variation, and given that this confidence interval, the bulk of it, is far less than the bulk of the confidence interval for Bank B, we can also surmise that there's typically going to be less wait time at Bank A then at Bank B. There's a slight overlap between the upper limit for Bank A and the lower limit for Bank B, but the bulk of the confidence interval for Bank A is less than that for Bank B. So we can say that there's going to be on average less variation for the waiting times in Bank A than for Bank B. So we would actually prefer to be in line in Bank A rather than Bank B.
Now let's scroll back down and look at our answer options. If we compare the different answer options, we can see that Answer Options A and B will not be correct because it says the multi-line system has a lower variation, and we just showed that it's a single-line system that has the lower variation.
So this leaves us with Answer Option C and D. The difference between Answer Option C and D is to select which of the two systems appears to be better. And as we saw by comparing the confidence intervals, the single-line system appears to be better. This leads us to select Answer Option C. Good job!
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