"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.

23. Points Earned Construct a 90% prediction interval for total points earned in Exercise 13 when the number of goals allowed by the team is 140."

Accepted Answer

Hello there. Today we're going to 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 company uses a regression equation Y hat equals 120 minus 0.8 multiplied by X to predict monthly expenses Y. Based on the number of units produced X for a sample of N equals 20 months, X bar equals 50, S subscript XX equals 500, and the standard error of estimates subscript Y is equal to 12, construct a 90% prediction interval for monthly expenses when 60 units are produced. Awesome. So it appears for this particular prompt we're asked to construct a 90% prediction interval for monthly expenses when 60 units are produced. 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 us read off our multiple choice answers to see what our final answer might be, noting that they're all written in. of parenthesis. So A is 60.201.83.809. B is 48.737, 95.263. C is 55.766, 84.247, and D is 58.154,81.854. Awesome. So our first step that we need to take in order to solve this problem is we need to take our regression equation for Y hat and plug in our value that we know for X. So Y hat is equal to 10. 20 minus 0.8 multiplied by X, but we're told that X is equal to 60 in this case. So when we plug in a value of 60 in for x and we plug in our expression into our calculator, we will find that it's simply equal to 72. Our second step is we need to compute the value for S predicted, which is SP for short, which, as we should recall, this equation. Should be as follows plugging in our known variables. 12 multiplied by the square root of 1 plus, so we need to take 1. Plus 1 divided by 20 plus parentheses 60 minus 50 to the power of 2. Divided by 500. Fantastic. So when we plug in this expression into our calculator, we will find that it will be equal to drumroll, 13.416. Fantastic. So, with that in mind, since we now know that our value for SP, which is the S predicted, Which is going to be equal to 13.416. Our next step now is we need to determine the degrees of freedom, which, as we should recall, we can determine the degrees of freedom as, which we'll call DF for short, is equal to 20, minus 2, which is equal to 18. And we need to note that for a 90% prediction interval, our value for T or T to the power, I should say. So T to the power of star will be equal to based on The information provided to us by this particular prompt, it'll be equal to 1.734, and we can find this value using our tea table, which you should have a tea table in your textbook. If not, you can just quickly look one up online. So now our next, our next step now that we determine the degrees of freedom and our value for T to the power of star, we now need to determine the margin of error E, and the margin of error E is equal to 1.734 multiplied by. 13.416. So with that in mind, we need to take 1.734 multiplied by 13.416, which will be equal to, or rather, to be more precise, approximately equal to 23.263. When we round to three decimal places to match our multiple choice answers because they're all rounded to three decimal places. So we're trying to match up to our multiple choice answers, so we have to make sure we're rounding the same way. So, our next step, which is our fifth and final step is we need to create our prediction interval. So we need to take 72 plus or minus 23.263. Which is equal to parentheses 48.737, 95.263. And that's it. That is our final answer for the interval. Once again, making sure we round it to three decimal places, and that's it. So looking at our multiple choice answers, it corresponds to multiple choice answer letter B. Which once again is parentheses 48.737, 95.263. 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.

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.23. Points Earned Construct a 90% prediction interval for total points earned in Exercise 13 when the number of goals allowed by the team is 140."

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"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.
23. Points Earned Construct a 90% prediction interval for total points earned in Exercise 13 when the number of goals allowed by the team is 140."