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

Question

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

20. Construct a 90% prediction interval for the average time adults ages 35 to 44 spend per day watching television in Exercise 10 when the average time adults ages 25 to 34 spend per day watching television is 2.25 hours."

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 regression equation for predicting the average number of hours spent exercising per week by adults ages 50 to 59 from the averages for ages 40 to 49 X. Is Y hat equals 2.5 plus 0.65 X. The sample size is N equals 10, X bar equals 3.7. S X X is equal to 5.4 and S is equal to 0.5. Construct a 90% prediction interval for Y when X equals 4.2. Awesome. So it appears for this particular problem, we're asked to take all the information that is provided to us by the prom itself, and we're asked to construct a 90% prediction interval for Y when X is equal to 4.2. So now that we know that we're ultimately trying to determine this interval value for this particular prompt, Let's read off our multiple choice answers to see what our final answer for the interval might be, noting that they're all written inside parentheses. So A is 4.09.5.41, B is 4.00. 5.50. C is 4.23.6.23. D is 3.95.5.55. Awesome. So our first step that we need to take in order to solve this problem is we need to note that we're told that our regression equation is for this particular problem, so the regression equation states that Y hat. Is equal to 2.5 plus 0.65. X, but the value for X in this particular case is 4.2, so we need to plug in our value of 4.2. So when we take 2.5 plus 0.65 multiplied by 4.2, you will find that this is equal to 5.23. So that's our first step is we need to solve for our value for Y given a value X is equal to 4.2. Our second step now is we need to solve for SE subscript prediction. Which is equal to 0.5 multiplied by the square root of 1 + 1 divided by 10 plus parentheses 4.2 minus 3.7 to the power of 2, divided by 5.4. Awesome. So, moving right along here, when we plug in this expression into our calculator, we will find that SE prediction is approximately equal to, once again, rounding to 4 decial places, it's approximately equal to 0.5353. And this is when we take 0.5 multiplied by the square root of 1 + 1 divided by 10 + 4.2 minus 3.7 to the power 2 divided by 5.4, which is approximately equal to 0.5353. Our Next step now. is we need to. Determine the degrees of freedom. So that's our third step. So our third step now is determining the degrees of freedom, which we need to take 10 minus 2, which is equal to 8. So our degrees of freedom is equal to 8, which we'll call DF or short degrees of freedom DF. And we need to note that for a 90% confidence interval, we need to note that T to the power of star will be equal to 1.860. So now our next step is we need to determine our margin of error. So our margin of error, ME. is going to be equal to, so ME, our margin of error will be equal to 1.860, multiplied by 0.5353. Which will give us an answer of approximately. 0.9957. Fantastic. So now we need to determine our, that's this is our fourth step. We now need to determine our prediction interval, so we need to take 5.23. And we need to plus or minus our value of 0.9957. And when we do, we will find that it is equal to Parentheses. 4.234. Oh. 6.2257, which when we round to two decile places to match our multiple choice answers, we will find that it's approximately equal to parentheses 4.23.6.23. And that's it. That is our final answer. Hooray, we did it. And this corresponds to multiple choice answer letter C, which once again is parentheses 4.23, 6.23. 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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