In Exercises 5–8, use (a) randomization and (b) bootstrapping ...

Question

In Exercises 5–8, use (a) randomization and (b) bootstrapping for the indicated exercise from Section 9-1. Compare the results to those obtained in the original exercise.

Exercise 9 in Section 9-1 “Cell Phones and Handedness”

Accepted Answer

All right, hello, everyone. So this question says, a survey is conducted to test whether the proportion of employees who prefer remote work is lower among those in manufacturing compared to those in technology. Out of 210 manufacturing employees, 98 prefer remote work. Of out of 390 technology employees, 312 prefer remote work. At a 0.01 significance level, use randomization and bootstrapping to test the claim that the remote work preference rate is lower in manufacturing than in technology. All right, so first and foremost, let's talk about the data that we do have in this moment, right? We can distinguish between two distinct groups. We've got manufacturing or group M and technology or Group T. So first we have the sample sizes for both groups. For manufacturing, that's a total of 210 employees. And for technology, that's a total of, let's see, 390. And so from here, that's N for both groups. Now we can also find X, which in this context is the number of people in each group that prefer remote work. So XM for manufacturing is equal to 98, and XT for technology is equal to 312. Now we can calculate the sample proportions or Phat for both groups. So Phat, if you recall, is equal to X divided by N. So for manufacturing, for example, that's 98. Divided by 210. Which gives you approximately 0.4667. And so Phe of T is equal to 312, divided by 390. Which is equal to 0.80. And so because we have both values for Phat, we can also find the difference in both proportions. So in this case, that would be 0.80. Subtracted by 0.4667, which gives you 0.3333. Now We can distinguish between the null and alternative hypothesis hypotheses rather. The null hypothesis, that's H not, would state that PF M is equal to P T. And so what that means is that there's no difference in the remote work preference between both groups. The alternative hypothesis, on the other hand, or H1, would state that PFM is less than PFT. Meaning that manufacturing has a lower preference rate than technology. So now let's begin with randomization. And for randomization, that's the first approach, we have to pull the data together. So first, let me scroll down to open up some space. The first thing we need are the total number of employees, which in this case would be the sum of 210 and 390, which gives you 600. And now we can find the total number of employees that prefer remote work among both groups. So that's the sum of 98 and 312, which gives you 410. And then We can also find the number of people that do not prefer remote work. So in this case, that would be 600, subtracted by 410, which gives you 190 people not preferring remote work. So, for the randomization test, our next step is to reshuffle under the null hypothesis. And so what that means is that 210 employees are randomly assigned to manufacturing, and 390 are assigned to technology. Keeping the same total successes. Once those employees have been randomly assigned, what you then do. I compute the difference in proportions, that is Pha of T subtracted by Pha of M. For each reshuffled sample. And then you repeat this many, many times, upwards of 10,000 iterations. This then leads you to the distribution of differences. Under the null hypothesis. So With this scenario in mind, what result do we expect to see? Well, first of all, recall that after the distribution of differences is made, you would then compute the p-value, that is the proportion of permutated differences. So, if P is less than or equal to 0.01, you would then reject the null hypothesis. So, in this case, the permutated differences will rarely be as large as 0.3333. And that's because the observed difference happens to be quite large. So In this context, we would expect for P. To be approximately equal to 0.000. Because this is less than 0.01. We can reject the null hypothesis. So now, let's move on to bootstrapping. And so for boots bootstrapping, the first step is to resample with replacement. So from the original manufacturing data, you draw as many bootstrap samples, upwards of 10,000, with a size of 210, and calculate Pha of M. And then you would do the same thing for technology, this time with the sample size being 390. Now for each bootstrap sample. You would once again calculate Pha of T subtracted by Pha of M. And then you would calculate a 99% confidence interval or the true difference. Now, this is a one-tailed test, so you want to check if the lower bound of the confidence interval is greater than 0. So now, what result do you expect to see? Well, in this context, the 99% confidence interval would likely be something like 0.25. To 0.42. Because the entire interval is above 0. We would reject the null hypothesis as well. So now this leads us to the final answer, which reads that both methods provide statistically significant evidence that remote work preference is lower in manufacturing than in technology. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

In Exercises 5–8, use (a) randomization and (b) bootstrapping for the indicated exercise from Section 9-1. Compare the results to those obtained in the original exercise.

Exercise 9 in Section 9-1 “Cell Phones and Handedness”

Key Concepts

Randomization

Bootstrapping

Comparative Analysis

Watch next