"Constructing and Interpreting a Prediction Interval In Exerci...

Question

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.

28. Total Assets Construct a 90% prediction interval for the total assets in federal defined benefit plans in Exercise 18 when the total assets in IRAs are $6400 billion."

Accepted Answer

Hello, everyone, let's take a look at this question together. A biologist uses the regression equation Y hat equals 2.5 plus 1.2 X to predict the weight of a plant where Y is in grams based on the number of weeks grown, which is X. The standard error of the estimate is equal to 0.8. The mean number of weeks is equal to. And our sample size N is equal to 10, as well as our sum of the square difference between our X values and our mean is equal to 30, and we must construct a 99% prediction interval for a plant grown for 9 weeks. Is it answer choice A 12.431 to 14.443? Answer choice B, 10.512 to 13.523. Answer choice C, 10.123 to 16.477, or answer choice D 13.045 to 15.054. So, in order to solve this question, we have to recall how to construct a 99% prediction interval, so that we can construct one for a plant grown for 9 weeks, given our regression equation of Y hat equals 2.5 plus 1.2 X, where Y hat is the predicted weight of the plant in grams, and the number of weeks grown is given as X, and we are provided the standard error of the estimate, which is 0.8. The mean number of weeks, which is 6, the sample size is 10, and the sum of the squared difference between the X values and the mean is equal to 30. And so the first step in solving this problem is computing the predicted value, which is given as Y hat is equal to 2.5 plus 1.2. Multiplied by 9, as the value for X is the number of weeks grown, which in this case, it is 9 weeks, and this can be simplified into 2.5 plus 10.8, which is equal to 13.3. So our predicted value of Y hat is equal to 13.3, and then we have to compute the standard error of the predicted value. Which is given as 0.8, multiplied by the square root of 1 plus 1 divided by 10, plus the square difference of 9 and 6 divided by 30, and this formula can be simplified to give us 0.8, multiplied by 1.1832, which is equal to 0.9466, as the value for the standard error of the predicted value. And then the next step is to find the critical T value for 99% confidence, and our N subtracted by 2, which equals 8 as the value for our degrees of freedom, and using a standard table, the critical T value is approximately 3.355. And then we can compute the margin of error, which is given as E is equal to 3.355 multiplied by 0.9466. Which is approximately 3.177. And so now we can construct our prediction interval, which is given as 13.3, subtracted by the margin of error, which is 3.177 to 13.3 plus 3.177. So our prediction interval is 10.123 to 16.477. And looking at our answer choices, we can identify that answer choice C is the correct answer. As answer choice C contains the correct prediction interval for a plant grown for 9 weeks, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

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