Cell Phone Radiation Here is a sample of measured radiation em...

Question

Cell Phone Radiation Here is a sample of measured radiation emissions (cW/kg) for cell phones (based on data from the Environmental Working Group): 38, 55, 86, 145. Here are ten bootstrap samples:

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a. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the population mean.

Accepted Answer

Hello there. Today we're going to 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. Here is a sample of temperature measurements in units of degrees Celsius taken from laboratory samples after a heating process. Curly brackets 22, 25, 28, 32. The 10 bootstrap samples based on sampling with replacement are curly brackets, 22, 25, 28, 32. Curly brackets 28, 322528. Curly brackets 25, 22, 3228. Curly brackets 22, 28, 32, 25. Curly brackets 28 25 25 32. Curly brackets 32 28 22 28. Curly brackets 25 32 28 22. Curly brackets 28 22 28 25. Curly brackets 322528 22. Curly brackets 28, 32, 22, 25. Using only these 10 bootstrap samples, construct an 80% confidence interval estimate for the true mean temperature of the laboratory samples. Awesome. So that's our angle. Our final answer. We're ultimately trying to solve for is we're asked to take these 10 bootstrap samples, and we're asked to create or construct an 80% confidence interval to help us estimate for the true mean temperature of these specific laboratory samples. So with that said, our first step that we need to take. we need to calculate the means of each bootstrap sample. So we're given 10 curly brackets worth of data points, and each of our curly brackets we're given 4 data points. So we need to find the mean of each of our 10 bootstrap samples. So I've gone ahead and created a little. I guess a formula chart for each of our samples. So this is our first step. So we have sample 1 mean through sample 10 mean. And as you can see, when we take the mean for each of our samples, and I'll go in numerical order from sample 1 to sample 10, so that means going from the first set of curly brackets to the last curly set of brackets in order. Our sample one will be equal to 26.75, sample 2 would be equal to 28.25, sample 3 would be equal to 26.75. Sample 4 would be equal to 26.75, sample 5 would be equal to 27.50, sample 6, 27.50, sample 7 will be 26.75, sample 8 would be 25.75. Sample 9 would be 26.75, and then sample 10 would be equal to 26.75. And note the take the mean, we just add up our data points and then we divide by the number of data points we have, so we add the 4 data points together, and then we divide by 4. Awesome. So that's our first step. So our first step was to calculate the means of our bootstrap sample. Our second step now is we need to put our data points in ascending order. We need to order our means, so we need to take and write them in ascending order, so we need to take our mean values. So we have 25.75, 26. 75 and once again we're putting them in numerical order, ascending order, so from smallest to largest, so. Our next value would be 26.75, 26.75. So there's gonna be 12345, so 5, so 123, 26.75. So there's gonna be 26.75 5 times, so 26.75. Because as you can see in our data points we have at 12345. 6, actually 6 times. 12345. Yes. So, right, so let's check. So, 123456, so there's 6 of them. So then we need to take 26.75. So it's triple check once more, so 123456. It's important when you're doing any sort of methodology like this just to make sure that you're counting all the right numbers, make sure you write them all down so you don't accidentally leave a number out. And then after 26.75, our next number is 27. 50, and then we're gonna have another 27.50, and then we'll have 2 28.25s values, so 228.50. So. 28. 50 and then 28.25. OK, so 28.25 and 28.25. So, our third step now that we've put our means in ascending order. Our next step now is we need to note and recall that for an 80% confidence interval, we need to trim 10% in each tail. So therefore, as a result, we will use the 10th. And 90th percentiles of these 10 values, so P10 will be the second value. And it will be equal to 26.75 and P90 will be the ninth number. And this will be equal to 28.25. So therefore, that will mean that for an 80% confidence interval, our final answer without a doubt will have to be 26.75, and don't forget our units of measurement, which is degrees Celsius, so 26.75 °C will be less than mu, where mu will be our mean. And me is less than 28.25 °C, and that is our final answer. Hooray, we did it. 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

Population Mean

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