Testing Claims About Variation
In Exercises 5–16, test t...

Question

Testing Claims About Variation

In Exercises 5–16, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population.

Bank Lines The Jefferson Valley Bank once had a separate customer waiting line at each teller window, but it now has a single waiting line that feeds the teller windows as vacancies occur. The standard deviation of customer waiting times with the old multiple-line configuration was 1.8 min. Listed below is a simple random sample of waiting times (minutes) with the single waiting line. Use a 0.05 significance level to test the claim that with a single waiting line, the waiting times have a standard deviation less than 1.8 min. What improvement occurred when banks changed from multiple waiting lines to a single waiting line?

6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7

Accepted Answer

Hello. In this video, we are told that a meal delivery company has a standard deviation of 2.5 minutes in delivery times under its old logistics system. After switching to a new route optimization algorithm, the company recorded the following delivery times and minutes for the simple random samples of 12 deliveries. At the 0.05 significance level, we want to test the claim that the new, the new system results in less variation in delivery times than the old system. OK, so let's just go ahead and do a hypothesis test for this problem. So, the first thing we want to do is we want to set up our null and alternate hypothesis. Now, in this case here, because we're trying to determine that the claim that is the new system results in less variation in delivery times than the old system, then the no hypothesis is that the standard deviation. Must be greater than or equal to 25, and the alternate hypothesis states the following, which is that the standard deviation will be less than 25 or 2.5. So these are going to be hypothesis for the hypothesis test. Now, let's go ahead and start by finding the sample standard deviation. What we want to do is you want to go ahead and first take the mean. Of all the data from the sample, that means is going to be labeled as X bar, and the sum of all these points divided by the sample size. is going to be 31.1583. Since we have the mean, we will go ahead and calculate the sample variance, and the sample variance, which is the sum of all the points, minus. The mean squared. is going to be 7.3291. And so the standard error in this case, or the sample standard deviation. Is going to be the square root. Of 7.3291 divided by N minus 1. Now, in this case here, this is going to be the square root of 7.3291, divided. Divided by 11. And this will give us a sample standard error of 0.8163. Since we have the standard error, we are going to use a G2 test. That is going to be T2 equal to N minus 1 times the variance divided by the standard error squared. Or divided by our original value of sigma squared. So, this is going to be 12 minus 1. Multiplied by the standard deviation, which is 0.8163 squared, divided by 2.5 squared. And this will simplify to give us a G square uh test statistic of 1.173. And now we need to determine the critical value. For the critical value, we have a degree of freedom equal to N minus 1, which will be 11, and we have a significance level of 0.05. So, G squared. Of 0.05 with respect to the degree. A freedom of 11 is equal to 4.575. Notice that our test statistic of 1.173 is less than 4.575. What this means is that we reject The null hypothesis. And the solution to this problem is going to be B. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Hypothesis Testing

P-value

Standard Deviation

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