"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.

27. Natural Gas Construct a 95% prediction interval for the export of natural gas from the United States in Exercise 17 when the marketed production of natural gas in the United States is 31 trillion cubic feet."

Accepted Answer

Hello 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. An economist uses the regression equation Y equals 12,000 plus 250 multiplied by X to predict annual income Y in dollars based on years of education X, the standard error of the estimate is S subscript E is equal to 4000. The mean years of education is X bar is equal to 14. N is equal to 20 and the sum of parentheses X subscript i minus x 2 is equal to 80. construct a 99% prediction interval for a person with 18 years of education. Awesome. So it appears for this particular problem, where asked to determine rather, construct a 99% prediction interval for a person with an 18 years of education. So a person with 18 years of education. So now that we know that we're ultimately trying to determine what this interval value is based on all the information that is provided to us by the prom itself, let's read off our multiple choice answers to see what our final answer might be, noting that they're all written inside a parenthesis to denote the interval. So A is 5,316, 86,954. B is 3630, 29,370. C is 23,000, 33,000. D is 20,000, 36,000. Awesome. So, our first step that we're gonna need to take in order to solve this particular problem is we need to compute the predicted value Y had, and Y is equal to 12,000 plus 250 multiplied by our value of X, which are value for X for this particular problem will be 18, means we're told that the person has 18 years of education. So we need to plug in our value of 18 in for X, and when we do, and we plug in this expression into our calculator, we will find that it's simply equal to 16,500. Fantastic. So now, our second step now is we need to compute the value for S predicted, which I'll call SP for short, which, as we should recall, plugging in our known variables as we go. The equation will be SP is equal to 4000 multiplied by the square root of 1 + 1 divided by 20 plus parentheses 18 minus 14. Of 2, divided by 80, and when we plug in this expression into our calculator and we round to the nearest whole number, we will find that it's equal to 4,472. Our third step now is we need to find the criticalt value for a 99% confidence, and we need to note that N minus 2 is equal to 18 degrees of freedom, which I'll call DF for short. So that will mean that our critical value for T subscript 0.005, 18 will be approximately equal to 2.878. Fantastic. Our fourth step now is we need to compute the margin of error E, and the margin of error is equal to 2.878, multiplied by 4,472. Which will be approximately equal to when we rounded the nearest whole number, 12,870. Fantastic. So now we need to construct our prediction interval, which is our 5th and final step, so we need to take parentheses. 16,500. Minus. 12,870, 16,500 plus 12,870. And when we performed this. Expression, we will find that it's equal to parentheses 3630, 29,370, and that is our final answer for the interval. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be multiple choice answer letter B, which once again is parentheses 3630, 29,370. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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