Comparing Waiting Lines

The values listed bel...

Question

Comparing Waiting Lines

The values listed below are waiting times (in minutes) of customers at the Jefferson Valley Bank, where customers enter a single waiting line that feeds three teller windows. Construct a 95% confidence interval for the population standard deviation sigma.

Waiting times in minutes: 6.5, 6.6, 6.7, 6.8, 7.1, 7.3, 7.4, 7.7, 7.7, 7.7.

Accepted Answer

A chemist records the reaction times in seconds for two different catalysts. Construct a 95% confidence interval for the population variance for each catalyst. Based on your results, is there a significant difference in variation, and we're given two data sets with catalyst A and Catalyst B. Now to solve this, we'll make use of a chi squared confidence interval. But we first need to find our means and variants of both data sets. So for I mean, we'll first sum all our values together. And divide them by the sample sizes. So you have 8.2 plus 8.4 plus 8.3 plus 8.1. Plus 8.5 divided. By 5 That means, gives us a value of 8.3 seconds. We now have our variants. The variance is given by the sum. Of our observation Minus the mean. Squared divided by sample size minus 1. And this will change based on which data set we're using. In this case, for our standard deviation, or our variance in this case, the standard deviation squared. We will use Our deviations first. We have 8.2 as our first value. Minus 8.3. Squared, which gives us 0.01. We have to do that for all of our values. We repeat this, we have 0.01, 0.01, 0, 0.04, and 0.04. If we add all of them together, We get 0.10 divided by 5 minus 1 for a sample size minus 1. This is 0.10 divided by 4, which is 0.025 seconds. We now have mean and er deviation for our first data set. For the other data set, we will repeat the process. Since it's the same process, I will just write down the answer of 8.1 seconds for the mean. And for the variance. We have 0.025 seconds. And that is seconds squared for the variance. Now, let's use the formula for confidence interval. Our confidence interval formula. It's given by. And minus 1 multiplied by variants. Divided by Archive squared statistic. We have chi square of alpha divided by 2 with degrees of freedom of N minus 1. And for the upper one, We have N minus 1 multiplied by variants. Divided By chi squad of 1 minus alpha divided by 2. And minus 1 degrees of freedom. And so we noticed then our alpha will be. 0.05, meaning alpha. Divided by 2 is 0.025. This means looking for the chi square statistic. Of 0.025 with degrees of freedom. Of 4. And a chi squared statistic of 0.975 degrees of freedom of 4. We can find these from the table. This gives us 11.143. And 0.484 for our values for chi square. We can plug into our confidence interval. A confi interval for a. Is 5 minus 1 multiplied by the variance of 0.025. Divided by Our 1 143 And in our upper bound is the same thing, 5 minus 1 multiplied by 0.025. All divided by 0.484. This gives us two values for our first one. 0.0090. 0.2066. In seconds. And we notice that conflict interval B will be the same. Since we have the same values or standard deviation and variants and Chi square We get 0.0090, 0.2066 seconds. And these are variances, so actually it's 2 squared. So now we have our 2 confidence intervals. And they're actually the same confidence interval. Now, we'll use a test to see if the variants differ. We'll use an F test to test this. And we'll say variance. F is equals to the first variants divided by the second variance. Which in this case, is 0.025, divided by 0.025, which is just 1. And now we need to look at the critical value. F critical, Is given by Alpha Divided by 2. Degrees of freedom of the first group and degrees of freedom of the second group. This means our F critical is gonna be for 0.025 with degrees of freedom of 4 and 4. From the table, this gives us 9.60. Now this is our upper critical value. Our lower value will be the F critical of 1 minus alpha divided by 2. With degrees of freedom, both of 4. So you have F critical, 0.975, 4 and 4. And We can find this value on the table, but we end up getting 0.104. Now 0.104. is less than our value of 1, which is less than the value of 9.60. This is inside of our range, which means we will fail to reject our hypothesis. Now, because we failed to reject the hypothesis, we will then say that there is no statistical evidence. That Variability. Differs between the groups. And that is our final answer to the problem. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Comparing Waiting Lines

The values listed below are waiting times (in minutes) of customers at the Jefferson Valley Bank, where customers enter a single waiting line that feeds three teller windows. Construct a 95% confidence interval for the population standard deviation sigma.

Key Concepts

Confidence Interval

Standard Deviation

Chi-Square Distribution

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