"In Exercises 19-24, construct the indicated prediction interv...

Question

"In Exercises 19-24, construct the indicated prediction interval and interpret the results.

22. Construct a 95% prediction interval for the fuel efficiency of an automobile in Exercise 12 that has an engine displacement of 265 cubic inches."

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. A study models the relationship between the number of hours studied X and exam score Y using the regression equation Y hat is equal to 55 + 3.2 multiplied by X. The sample size is N equals 20. The mean number of hours is X is equal to 10. The sum of parentheses X subscript I minus X 2 is equal to 180, and the standard error of estimate is S equals 4.1. Construct a 95% prediction interval for a student who studied X subscript 0 equals 14 hours. Fantastic. So it appears for this particular prom, we're asked to construct a 95% prediction interval for a student who studied X subscript zero or X0 for short, is equal to 14 hours. So now that we know that we're ultimately trying to determine what this interval value is. Based on the information that is provided to us by the prom itself, let's read off our multiple choice answers, noting that they're all written inside a parentheses because they're written in interval notation. A is 98.20,106.61. B is 97.85, 107.01. C is 96.45, 104.29, and finally D is 90.61,108.99. OK, so our first step that we need to take is we need to compute the predicted value. So we need to use our regression equation. Y hat is equal to 55 + 3.2. And we need to note that 3.2 multiplied by X, but we're told that X in this particular case is equal to 14 hours, so we need to plug in a value of 14 in for X. So when we take 55 plus 3.2 multiplied by 14, we will find that it's simply equal to 99.8. Our second step now is we need to compute the value of S predicted, which we'll call SP for short, which is equal to 4.1 multiplied by the square root of 1 + 1 divided by 20 plus parentheses 14 minus 10 2 divided by 180. And when we plug this this expression into our calculator, we will find that it's approximately equal to, when we round to three decimal places, 4.375. So once again, it's approximately equal to 4.375. And with that in mind, our third step now is we need to find the T value for a 95% confidence and noting that we have Uh degrees of freedom that's equal to 18, so our DF degrees of freedom is equal to 18. So that will mean that our T value, which is T subscript 0.025.18 is equal to 2.101. Fantastic. And once again, that was to help us find our T value. So our fourth step now is we need to compute the margin of error, which we'll call ME for short, and the margin of error is equal to 2.101, multiplied by 4.375, which is equal to 9.19 when we round to two decimal places. And our fifth and final step is we need to construct our prediction interval. So we need to take 99.8, and we need to take this value of 99.8, and we need to plus or minus. 9.19. So that means ultimately what this plus and minus symbol means in case you're unfamiliar with it, means we're going to take 99.8 plus 9.19 and so we have to do plus, and we also have to take 99.8 minus 9.1. And when we do, we will find when we write an interval notation that it will be equal to once again rounding the two decimal places to match your multiple choice answers. We will find that it's equal to parentheses 90.61, 108.99, and that's it. That's our final answer for the interval. Hooray, we did it, and this corresponds to multiple choice sensor letter D, which once again is parentheses 90.61, 108.99. And that's it. That's our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

"In Exercises 19-24, construct the indicated prediction interval and interpret the results.
22. Construct a 95% prediction interval for the fuel efficiency of an automobile in Exercise 12 that has an engine displacement of 265 cubic inches."

Key Concepts

Prediction Interval

Regression Analysis

Confidence Level and Interpretation

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