McDonald’s versus Wendy’s A student wanted to determine whethe...

Question

McDonald’s versus Wendy’s A student wanted to determine whether the wait time in the drive-thru at McDonald’s differed from that at Wendy’s. She used a random sample of 30 cars at McDonald’s and 27 cars at Wendy’s and obtained these results:

c. Is there a difference in wait times at each restaurant’s drive-thru? Use the α=0.1 level of significance.

Table showing wait times in seconds for 27 cars at Wendy’s drive-thru, with values ranging from 71.02 to 471.62 seconds.

Note: The sample size for Wendy’s is less than 30. However, the data do not contain any outliers, so the Central Limit Theorem can be used.

Accepted Answer

Hello. In this video, we are told that a researcher wants to compare the average time it takes to process a loan application at two different banks. She collects a random sample of 35 applicants from Bank A and a sample of 22 applicants from Bank B, and the results are labeled in the given table. We want to determine if there is a significant difference in mean possessed processing times between the two banks. We want to use a significance level of 0.1 and perform a two sample T test, assuming unequal variances. So let's go ahead and determine the given variables. Well, before we do that, let's actually go ahead and determine our hypothesis for the two sample T tests. Now, the null hypothesis supports the original claim, and the original claim from the researcher is that the average time it takes to process a loan application from both the banks is going to be the same. So the no hypothesis. It is going to be that the average time from Bank A is equivalent to the average time of Bank B. Now, our alternate hypothesis states the opposite. Our alternate hypothesis will state that there is going to be a difference between the average processing times between both of the banks. So both of them mean average times are not going to equal to each other. Now for Bank A, we are given a sample size of 35, we have a sample mean for bank A. Actually, let's go ahead and do this. So, NA is going to equal to 35. Our sample mean for bank A, so X bar sub A, is going to equal to 5.8 days, and our sample standard deviation for A is going to equal to 1.7. And for Bank B, the sample size with respect to Bank B is going to be 22. Our sample mean for Bank B is going to be 4.9 days, and our sample standard deviation for bank B is going to be 1.3. And for both banks, we want to use a significance level of 0.1. So, the first thing we're going to do is we're going to determine a tea testing statistic for unequal variances. Now, for unequal variances, the Test statistic is defined as the following. T is going to equal to the difference of the mean from bank A minus the mean from Bank B, and this is going to be divided by our standard error. And in this case here, standard error is going to be defined as the square root. The square root of the sample standard deviation of bank A squared divided by the sample size from bank A. Plus The sample standard deviation from bank B2d divided by the sample size of bank B. Now, this is going to give us a standard deviation of the square root. Of 1.7 quantity squared divided by 35 plus 1.3 quantity squared divided by 422. And this is going to approximate to give us a sample standard deviation of 0.3992. We will set that to the site for now. Now, if we take this value and plug it back in to the Test statistic definition, since we know what the means between both of the banks are, this is going to be defined as 5.8 minus 4.9 divided by 0.3992. And by using a calculator, this value is going to approximate to give us a Test statistic of 2.255. Now that we have the Test statistic, the next thing we want to do is you want to go ahead and determine the degree of freedom. Now the degree of freedom is going to be defined as the following. It is the standard error squared divided by the following quantities. It is The standard deviation of bank A squared, divided by the sample size of bank A. Divided by the sample size of bank A minus 1, plus the sample standard deviation of bank B squared divided by the sample size of Bank B, divided by the sample size of bank B minus 1. Now, if we were to plug in the following values, the degree of freedom is going to be defined as the quantity 1.7 squared, divided by 35, plus 1.3 squared divided by 22, that entire quantity squared. And this will be divided. By 1.7 squared divided by 35, that entire quantity squared, and this will be divided by 35 minus 1, which is 34, and we will take the sum of 1.3 quantity squared, divided by 22, that entire quantity squared, divided by 1 less than the sample size of B, which is going to be 21. Now with the help of a calculator, this is going to give us an approximate degree of value. A degree of freedom of 52.76. But since we can only work with integer values for the degree of freedom, we are going to round up and say that the degree of freedom is going to be 53. So this is going to be the degree of freedom for the two-tail test. Finally, what we want to go ahead and do is we want to go ahead and determine the critical value. Now, in order to do this, we need to use a tea table. Now, because it's a two-tail test, we are going to divide our significance level by 2, and that is going to give us a significance level of 0.05. So, we are going to use a T table for the significance level of 0.05, a degree of freedom of 53, and we have a T test statistic of approximately 2.255. By using a T table, the T critical value we are going to compare our test statistic to is going to be approximately 1.675. Now, if we were to compare the Test statistic to the T critical value, the absolute value of the Test statistic is going to be greater than the T critical value. And so what that means is that we are going to reject our no hypothesis. And since we are going to reject our no hypothesis, we are going to accept the alternate hypothesis and state that there is a significant difference in the wait times between the two banks. So, the solution to this problem is going to be C. 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 for Two Independent Samples

Central Limit Theorem (CLT)

Significance Level and p-value

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