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

Question

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

21. Construct a 95% prediction interval for the number of hours of sleep for an adult in Exercise 11 who is 45 years old."

Accepted Answer

Below 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 study models the number of weekly exercise hours Y. As a function of age X, using the regression equation Y hat equals 6.5 minus 0.04 X. The standard error of estimate is S equals 0.6, sample size N equals 25, mean age X equals 35, and the sum of Xi minus X squared is equal to 800. Calculate the 95% prediction interval. For a 40 year old individual. Use T power a star is equal to 2.074 for 23 degrees of freedom. Awesome. So it appears for this particular problem, we're asked to take all the information that is provided to us by the problem itself, and we're asked to calculate the 95% prediction interval for a 40 year old individual. And now that we know that we're ultimately trying to determine what the value for the prediction interval is for this particular prompt, let's take a moment to read off our multiple choice answers to see what our final answer for the interval might be, noting that they're all inside of parentheses. So A is 4.87.6.02, B is 3.61, 6.19. C is 4.92.5.12 and D is 3.7 and 5.9. Awesome. So our first step that we need to take in order to solve this particular problem is we need to calculate the predicted value. So we need to take Y hat 0 is equal to 6.5 minus 0.04. And we need to multiply this because once again we're told that Y hat is equal to 6.5 minus 0.04 X, and we're told that our value for X in this particular case is 40, so we need to take 6.5 minus 0.04 multiplied by 40. And when we perform that calculation in our calculator, we will find that it's simply equal to 4.9. Our second step now is we need to compute the margin of error which will denote as ME, and we need to recall and use the prediction interval margin of error formula which states that ME, the margin of error, is equal to T to the power of star multiplied by S multiplied by the square root of 1 + 1 divided by N. Plus X 0 minus X to the power of 2 divided by the sum of Xi minus X to the power of 2. Fantastic. So now we need to plug in our known variables, so we know that X 0, so once again, we're plugging in our known variables, and we're told that the value of X0 minus X will be equal to 40 minus 35, which is simply equal to 5, and parentheses X0 minus X 2 will be equal to 25. And we're told that 25 divided by. 800 is equal to 0.03125, and 1 divided by N is equal to 1 divided by 25, which is equal to 0.04. So that will mean plugging in our known variables into our margin of error equation, you will find that ME is equal to 2.074 multiplied by 0.6, multiplied by the square root of 1 + 0.04 plus 0.03125. Which is approximately equal to let me round to two decimal places, 1.29. Fantastic. So now, we need to construct our prediction interval. So that's our third step. So, constructing our prediction interval, we will find that 4.9, so to be specific, so let's be specific. So once again we're constructing our prediction interval, so we need to take Y 0, which is plus or minus, as we should recall, the margin of error. So we take our regression or regression equation information. So we take Y 0 plus or minus our margin of error, which in this case will be equal to 4.9 plus or minus 1.29. Which will mean that our final answer, which for the interval will be parentheses 3.61.6.19. And that's it. That is our final answer. So once again we take our value of 4.9 and we add it to 1.29, and then we take 4.9 minus 1.29 and we will produce the following interval. So with that in mind, looking at our multiple choice answers, the correct and final answer without a doubt has to be multiple choice answer letter B, which once again is parentheses 3.616.19. In parentheses, 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.

"In Exercises 19-24, construct the indicated prediction interval and interpret the results.21. Construct a 95% prediction interval for the number of hours of sleep for an adult in Exercise 11 who is 45 years old."

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"In Exercises 19-24, construct the indicated prediction interval and interpret the results.
21. Construct a 95% prediction interval for the number of hours of sleep for an adult in Exercise 11 who is 45 years old."