Comparing Waiting Lines

The values listed bel...

Question

Comparing Waiting Lines

The values listed below are waiting times (in minutes) of customers at the Bank of Providence, where customers may enter any one of three different lines that have formed at three teller windows. Construct a 95% confidence interval for the population standard deviation sigma.

Customer waiting times at Bank of Providence: 4.2, 5.4, 5.8, 6.2, 6.7, 7.7, 7.7, 8.5, 9.3, 10.0 minutes.

Accepted Answer

Welcome back, everyone. In this problem, a sample of waiting times in minutes at a local pharmacy is 365475645 and 6. Construct a 95% confidence interval for the population standard deviation sigma. Now, how do we find this 95% confidence interval for the population standard deviation? We recall, OK, that it's going to be equal to. The square root of N minus 1 multiplied by the variance divided by or critical chi squared value from the right, OK, for the lower bound and the square root of N minus 1 multiplied by Squared divided by or critical chiquad value on the left. For the upper bone, OK, so notice here in both of these. We'll need our chi square values for both sides of our two-tailed test and we'll need the sample variants, OK? Where the sample variants. is equal to the sum of the squared deviations, OK? The sum of X I minus X2 divided by the sample size N minus 1. Now notice here we have each of these points, each of the X points in our sample, and we can use that to find the sample mean and thus use that to solve for our variant. When we find a variance, then we can use that to find the confidence interval for the population standard deviation. So let's start with the sample variance. Now, before we find it, we know that we will need the sample mean. So let's add all of the values up 3 + 6 + 5 + 4 + 7 + 5 + 6 + 4 + 5 plus 6, OK, all divided by our sample size N, which is 10. And now when we do that, we're gonna have 51 divided by 10, which equals 5.1. Now that we have the sample mean, we can try to find the sample variance. So, to make it easier, let's set up a table, OK, where we're gonna have our X values, Xi, OK? And then we're gonna have The difference between these values XI minus X bar, and then we're gonna square that. Because then when we square these differences, we can sum them to find the sample variants. So let's set up our table, OK? And in our table. put on a little bit more. Then we're gonna have each of our values. So that's 36, OK, uh, 5475, and then 645 and 6. Now we know our sample mean is 5.1. So for example, when X is 3, the difference is gonna be 3 minus 5.1, which equals -2.1. And 2.1 squared is 4.41. Likewise, for 6, it's gonna be 0.9, 0.9 squared is 0.81, for 5, negative 0.1, and that squared is 0.01. And if we continue downwards, for 4, it's gonna be -1.1 and 1.21. For 7, uh 1.9. And 3.61, or 5 or negative 0.1 and 0.01. 6 is 0.9 and 0.81. 4-1.1 and 1.21. 5 negative 0.1 and 0.01, and 6 is 0.9 and 0.81. Now that we have all of our squared uh differences, then we can go ahead and sum them, OK? So that, no, that means the sum of XI minus XY 2 will be equal to, when we add all of these up, 12.90, OK? Now that we have this sum of differences, that means our sample variance is going to be equal to. The sum of difference is 12.9 divided by n minus 1, N minus 1, which is 9. I know 12.9 divided by 9 will be approximately equal to 1.4333. OK? So, this is our sample variant. Now that we have the sample variants, let's try to figure out our critical chi squared values, and we can use our chi square table. Now, here Or Degrees of freedom equal to the sample size minus 1. That's 10 minus 1, which equals 9, OK. At a 95%. Confidence interval, OK. For a two-tailed test. OK. Then Or chi squared value, OK. Let me put that properly here. Our chi squared value for or tail area. Now remember we have a 5% area that's gonna be split into two. So for the first part of the tail, it would be 0.025 for the tail area 9 degrees of freedom. That chi squared value would be equal to 19.023. On the other hand, the cash create value for the next part of our uh distribution, OK. So that's 2.5% from 100, that's 0.975 and 9 degrees of freedom will be equal to 2.700. Know that we have these values and we have our variants, we can use them in our formula for the confidence interval. So, thus, let me put that in red here. Therefore, yeah. For our confidence interval. It's going to be the square root of 9 multiplied by 1.4333 divided by 19.023, OK, as the lower limit, and the square root of 9 multiplied by 1.4333 divided by 2.700 for the upper limit. And when we go ahead and solve, we get it to be equal to 0.823 for the lower limit and 2.186 for the upper limit. Thus, the 95% confidence interval for the population standard deviation is from 0.823 to 2.186 minutes. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Confidence Interval

Standard Deviation

Chi-Square Distribution

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Comparing Waiting LinesThe values listed below are waiting times (in minutes) of customers at the Bank of Providence, where customers may enter any one of three different lines that have formed at three teller windows. Construct a 95% confidence interval for the population standard deviation sigma.

Key Concepts

Confidence Interval

Standard Deviation

Chi-Square Distribution

Watch next