"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.

30. New Vehicle Sales Construct a 99% prediction interval for new vehicle sales for Honda in Exercise 20 when the number of new vehicles sold by Toyota is 2159 thousand."

Accepted Answer

Hello everyone. Let's take a look at this question together. A nutritionist predicts daily calorie intake Y based on hours of physical activity per week. X, using the regression equation Y hat equals 1800, subtracted by 50 X. The standard error of estimate is equal to 150, the sample size is equal to 25, the mean hours of activity is equal to 5, and the sum of the squares of X is equal to 250. Construct a 90% prediction interval for a person who exercises 8 hours per week. Is it answer choice A, 1,133.5 to 1,666.5? Answer choice B, 1,338.1 to 1,761.9. Answer choice C, 1400 to 1700, or answer choice D, 1,380.5 to 1,720.5. So in order to solve this question, we have to recall how to construct a 90% prediction interval so that we can construct one for a person who exercises 8 hours per week, given our regression equation, the standard error of estimate, the sample size, the mean hours of activity, and the sum of squares of X values. And so the first step in constructing our 90% prediction interval is to compute the predicted value. Using our regression equation, so Y hat is equal to 1800, subtracted by 50, which is multiplied by 8, as we are constructing a 90% prediction interval for a person who exercises 8 hours per week, so we substitute 8 for the variable X, which gives us 1800, subtracted by 400, so our predicted value is equal to 1400. And then we compute the standard error of the prediction. Which is equal to 150, multiplied by the square root of 1 plus 1 divided by 25, plus 8 subtracted by 5 squared, divided by 250. And when simplified, we have 150 multiplied by 1.0367, so our standard error of the prediction is equal to 155.505. And then we calculate our degrees of freedom, which is N subtracted by 2. So our sample size N, which is 25, subtracted by 2 is equal to 23, and using a T table for 90% confidence, our T critical value is equal to 1.714. And now we can compute. The margin of error, which is given as E, which is equal to 1.714, multiplied by 155.505, so our margin of error E is approximately 266.5. So now we can construct our prediction interval, which is given as 1400 plus and minus 266.5, so our prediction interval is equal to 1400, subtracted by 266.5 to 1400 plus 266.5. Or 1,133.5 to 1666.5, which looking at our answer choices, we can identify that answer choice A is the correct answer, as it contains the correct 90% prediction interval for a person who exercises 8 hours per week, and all other intervals are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

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