"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.

25. Mean Wage Construct a 99% prediction interval for the mean annual wage in Exercise 15 when the percentage of employment in STEM occupations is 13% in the industry."

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 researcher developed a regression model to predict the final exam score. based on the number of hours studied X, the regression equation is Y is equal to 50 + 2 multiplied by X. The sample size is n is equal to 16. The mean number of hours studied is X is equal to 10. S subscript XX is equal to 240, and the standard error of estimate is S subscript Y is equal to 5. Construct a 90% prediction interval for the final exam score when a student studies for 14 hours. Awesome. So it appears for this particular problem we're ultimately trying to create or construct a 90% prediction interval for the final exam score when a student studies for 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 problem itself, let's take a moment to read off our multiple choice answers to see what our final answer might be. Based on what we've learned from the problem itself and what we're trying to solve for and noting that they're all written inside a parentheses, A is 72.945, 87.065, B is 74.12485.880. C is 68.64487.356. And D is 70.506, 89.505. Awesome. So our first step that we need to take in order to solve this problem is we need to take our equation that's given to us by the problem itself for the regression equation for Y hat, and we need to take our known value for X and plug it into this equation. So Y hat is equal to 50 + 2 multiplied by our value of X, which X in this case is 14, because we're told that the student studies for 14 hours. So when we plug in this value for X and and write this expression into our calculator, and we calculate it, we will find that it is equal to simply 78. Our second step now is we need to compute, or rather solve for the value for S predicted, which, as we should recall the equation to solve for this particular value is as follows, and we're going to plug in our known variables as we go. So SP is equal to 5 multiplied by the square root of 1 + 1 divided by 16 plus parentheses 14 minus 10 to the power of 2. Divided by 240. Which this this expression when you plug it into your calculator will be approximately equal to 5.3131 when we round to 4 decimal places. Our third step now is we need to solve for the degrees of freedom, which we'll denote as DF and as we should recall for this particular problem, the degrees of freedom will be equal to 16 minus 2, which is equal to 14. And for a 90% prediction interval, the value for T power of star will be equal to 1.761, and we can solve for this variable by using our T table, or rather we need to recall how to use a tea table, and you should have a tea table in your textbook. If not, you can quickly look one up online. Our fourth step now is we need to determine the margin of error, which will denote as E, and the margin of error is equal to 1.761 multiplied by, so we need to multiply 1.761 multiplied by 5.3131. Which when we round to three decimal places, because all of our multiple choice answers are rounded to 3 decimal places, we will find that the margin of error is approximately equal to 9.356. So our fifth and final step now is we need to we need to write our expression for the prediction interval, which it will be 78 plus or minus 9.356, which will be equal to parenthesis. 68.644, 87.356 in parentheses, and that's it. That's our final answer for the interval. Hooray, we did it, and this corresponds to multiple choice answer letter C, which once again is parenthesis 68.64487.356. 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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