Seating Choice In a 3M Privacy Filters poll, respondents were ...

Question

Seating Choice In a 3M Privacy Filters poll, respondents were asked to identify their favorite seat when they fly, and the results include these responses: window, window, other, other. Letting “window” and letting “other”, those four responses can be represented as {1, 1, 0, 0}. Here are ten bootstrap samples for those responses: [Image]
Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the proportion of respondents who indicated their favorite seat is “window.”

Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the proportion of respondents who indicated their favorite seat is “window.”

Accepted Answer

Below there, today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A small startup surveys for employees about their favorite morning drink and records coffee, coffee, tea, tea. Coding coffee as 1 and T as 0 produces the data curly brackets of 1100. 10 bootstrap samples resampling with the replacement, each of size 4, yield these sorted bootstrap proportions of one's 0.00. 0.25, 0.25, 0.50, 0.50, 0.50, 0.75, 0.75. 0.75 1.00. Using these 10 values, construct an 80% confidence interval estimate of the true proportion of employees whose favorite drink is coffee. Awesome. So it appears for this particular prompt, we're asked to use these 10 bootstrap values to help construct or create an 80% confidence interval to estimate the true proportion of the employees whose favorite drink is coffee. Awesome. So now that we know what we're ultimately trying to solve for, our first step is we need to make sure we sort our bootstrap proportions in ascending order, meaning from smallest to largest, and as you can see by our like from the problem itself, it already is organized in ascending order, meaning it's from the smallest number to the largest number. So that means now our next step is we need to, since now we have it in ascending order, so that's a checkmark, our second step is we need to trim 10% at each tail, meaning we need to take alpha divided by 2, where alpha denotes the significance. So alpha divided by 2, and alpha divided by 2 in this case is equal to 0.10 or 10%. So now, we need to drop the lowest 10% of values, meaning we need to remove the first value, which is 0.00. And once again, this is our first value. Fantastic. So once again, that's our first value, and then we need to drop this highest, so now we're dropping the highest 10% of values. So that means we need to remove the 10th value, which is our 1.00. Once again, this is our 10th value. So now, our third step is we need to focus our attention on the remaining values. So this is positions 2 through 9. So that means we're focusing our attention on 0.25, 0.25, 0.50, 0.50, 0.50, 0.75. 0.75 and finally, our last value which is 0.75. OK. So now our fourth and final, actually rather our fourth step, means we're almost close to the end, but not quite, to solving this problem. So now we need to focus our attention on the bounds of the 80% confidence interval. So, let us note that our lower bound, which I'll call LB for short, so our lower bound. is equal to 0.25, and this is our second smallest number and our upper bound, which I'll call UB, and our upper bound is equal to 0.75, which is our ninth smallest number. So that will mean that our fifth and final step means that for an 80% confidence interval, that would mean that 25%. Is less than P. And P is less than 75% where P denotes our proportion. And that's it. That is our final answer. Hooray, we did it. So thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Bootstrap Sampling

Confidence Interval

Proportion

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