"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.

24. Trees Construct a 90% prediction interval for the trunk diameter of a tree in Exercise 14 when the height is 80 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. A nutritionist predicts daily calorie intake Y using the regression equation Y hat equals 1800 plus 15 multiplied by X, where X is the number of minutes of exercise per day for a sample of N equals 25, X equals 30, S subscript XX equals 600, and S subscript Y equals 40. Construct a 90% prediction interval for the calorie intake when a person exercises 40 minutes per day. Awesome. So it appears for this particular problem we're asked to construct a 90% prediction interval for the calorie intake when a person exercises 40 minutes per day. 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 for 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 of parentheses. So A is 2,324.69, 2,475.31, B is 2,280.95, 2,419.16. C is 2,215.28, 2,484.82 and D is 2,257.60,2,442.49. Awesome. So our first step that we need to take in order to solve this problem is we need to compute or determine the predicted value using our regression equation Y hat and plugging in our value for X. So Y hat is equal to 1800 plus 15 multiplied by our value of X, which in this case we're told that X is equal to 40 because they're exercising 40 minutes per day. So when we plug in this expression into our calculator, we will find that it's equal to 2400. And moving right along here, our second step is we need to compute the value for S predicted, which I'll call SP for short, which, as you should recall in plugging in our known variables, SP is equal to 40, multiplied by the square root of 1 + 1 divided by 25 plus parentheses 40 minus 30 to 2. Divided by 600. Which when we plug this into our calculator, we will find that it's approximately equal to 43.94. Fantastic. Our third step now is we need to determine the degrees of freedom, which we'll call DF for short, and the degrees of freedom, as we should recall, for this particular for this particular problem will be equal to 25 minus 2, which is equal to 23. And we need to note that for a 90% prediction interval, the value for T to the power of will be equal to 1.714, and we can find this value from our tea table, which you should have a tea table in your textbook. If not, you could quickly look one up online. Our next step now is we need to determine the margin of error E, and the margin of error is equal to 1.714 multiplied by 43.94. And when we multiply those two values together, we will find that it is approximately equal to 75.31 when we round to two decimal places to match our multiple choice answers. And our fifth and final step is we need to create our prediction interval, so we need to take 2400 plus or minus, so we need to take 2400 plus or minus our margin of error value, which was 75.31. And when we perform that calculation, we will find that it's equal to parentheses 2,324.69, 2,475.31. And that is our final answer for the interval. Hooray, we did it. And this corresponds to multiple choice answer letter A, which once again is parentheses 2,324.69, 2,475.31. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.24. Trees Construct a 90% prediction interval for the trunk diameter of a tree in Exercise 14 when the height is 80 feet."

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"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.
24. Trees Construct a 90% prediction interval for the trunk diameter of a tree in Exercise 14 when the height is 80 feet."