7. FRESHMAN 15 Here is a sample of amounts of...

Question

7. FRESHMAN 15 Here is a sample of amounts of weight change (kg) of college students in their freshman year (from Data Set 13 “Freshman 15” in Appendix B): 11, 3, 0, –2, where –2 represents a loss of 2 kg and positive values represent weight gained. Here are ten bootstrap samples:

{11, 11, 11, 0}, {11, –2, 0, 11}, {11, –2, 3, 0}, {3, –2, 0, 11}, {0, 0, 0, 3}, {3, –2, 3, –2}, {11, 3, –2, 0}, {–2, 3, –2, 3}, {–2, 0, –2, 3}, {3, 11, 11, 11}.

b. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the standard deviation of the weight changes for the population.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says a study records the monthly water consumption in liters of households in a neighborhood. A small sample contains the values 150, 180, 170, and 160. 10 bootstrap samples are generated from this data. And we're given for bootstrap samples. For the first one, we have 150, 150, 180, and 180. For the next one, we have 170, 160, 180, and 160. For the next one, we have 160, 180, 170, and 150. For the next one, we have 150, 160, 160, and 160. For the next one, we have 180, 180, 180, and 170. For the next one, we have 150, 170, 150, and 170. For the next one, we have 160, 150, 170, and 150. For the next one we have 170, 160, 10. 60 and 170. And for the next one, we have 160, 160, 160, and 160. And for the last one, we have 180, 170, 180, and 160. Using the same bootstrap samples, construct a 80% confidence interval estimate for the standard deviation of the monthly water consumption. So, the first thing we need to do is to calculate the standard deviation for each of our bootstrap samples. So, we call the standard deviation S is equal to the square root of the quantity of the sum. Of the quantity of Xi minus X in quantity squared divided by the quantity of n minus 1, where X of by is going to be an individual data point in one of our samples, X bar is going to be the mean of that sample and N is going to be the number of data points in the sample, which in this case is 4 for all of them. And so what we need to do is calculate the standard deviation for each of our samples, and this is best done in a table. And so, in our table, for each row, we're gonna have one of our samples, and then we'll calculate the mean for that sample, as well as the standard deviation. So, for our first sample, Which is the 150, 150, 180, and 180. That has a mean of 165. And when we calculate the standard deviation, we get the square root of 300, which is approximately equal to 17.3. For a second sample, we have the values of 170, 160, 180, and 160. That gives us a mean of 167.5. And calculating the standard deviation using our formula, we get the square root of 91.67, which is approximately equal to 9.6. For our third sample, we have the values of 160, 180, 170, and 150, and that gives us a mean of 165, and a standard deviation of the square root of 166.67, which is approximately equal to 12.9. For the next sample, we have the values of 150, 160, 160, and 160. That gives us a mean of 157.5, and that gives us a standard deviation of the square root of 25, which is equal to 5. For our fifth sample, we have the values of 180, 180, 180, and 170, which gives us a mean of 177.5. And calculating our standard deviation, we get the square root of 25, which again, is equal to 5. For our sixth sample, we have 150, 170, 150, and 170, which gives us a mean of 160. And the standard deviation when we calculate it is the square root of 133.33, which is approximately equal to 11.5. For 1/7 sample, we have the values of 160, 150, 170, and 150. This gives us a mean of 157.5. And when we calculate the standard deviation, we get the square root of 91.67, which is approximately equal to 9.6. For our 8th sample, we have the values of 170, 160, 160, and 10,770, which is, uh, which gives us the mean of 165, and we calculate the standard deviation. We have the square root of 33.33, which is approximately equal to 5.6. For our ninth sample. We have the values of 160, 160, 160, and 160. That gives us a mean of 160, and a standard deviation of 0. And then for our final sample, we have the values of 180, 170, 180, and 160, which gives us a mean of 172.5, and we calculate the standard deviation, we get the square root of 91.67, which gives us approximately 9.6 for that value. So, in order to find our 80% um confidence interval, we first want to order these standard deviations in going from the smallest value to the largest value. So doing that, we will have 0. Then 5, then 5 again. Then 5.8. Then 9.6, then 9.6 again, then 9.6 for a third time. Then 11.5. Then 12.9, and then finally 17.3. So, for an 80% confidence level, we just need to keep 8 of our 10. Values here, so we're going to drop the Lowest value as well as the highest value. And then the remaining interval is going to be the estimate for a confidence interval. So, that means that our interval is going to be from 5 to 12.9, and that is going to have units of liters, since that is the units that our water was measured in. So this here is going to be the answer for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Bootstrap Sampling

Confidence Interval

Standard Deviation

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