Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Refer to exercise $21,$ where data on the production volume $x$ and total cost $y$ for a particu-lar manufacturing operation were used to develop the estimated regression equation $\hat{y}=$$1246.67+7.6 x .$$$\begin{array}{l}{\text { a. The company's production schedule shows that } 500 \text { units must be produced next }} \\ {\text { month. What is the point estimate of the mean total cost for next month? }} \\ {\text { b. Develop a } 99 \% \text { prediction interval for the total cost for next month. }}\end{array}$$$$\begin{array}{l}{\text { c. If an accounting cost report at the end of next month shows that the actual production }} \\ {\text { cost during the month was } \$ 6000, \text { should managers be concerned about incurring such }} \\ {\text { a high total cost for the month? Discuss. }}\end{array}$$

See explanation

Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

University of North Carolina at Chapel Hill

Piedmont College

Oregon State University

Boston College

Lectures

0:00

06:25

An important application o…

this problem. We're getting the following data set everything in black. And we're also given that, uh, the expected regression equation for this problem. Is that why I had equals 12? 46.67 plus 7.6 x. So I went ahead and calculated all the why hats all the predicted. Why values for each of the given x values, um, and will use this later on. The first thing we're asked to do is come up with a point estimate. Uh, that shows 500 units must be produced next month. Or what, The, uh, point estimate for the mean total cost for next month would be if 500 units must be produced. So what that means is that our why hat is going to be dependent on X equals 500. So if we go back to our regression equation, that would be 12 46.67 plus 7.6 times 500 and with this, we get the value of 5046.67 So this is our point estimate. And now we're asked to come up with a 99% ah prediction interval. So 95. A prediction interval is equal to the point. Estimate. Why? Hat Star, Plus or minus T at the Alfa over to the T test statistic. Adolf. Over too times, um, the center deviation of the independent value. So, uh, let's first calculate this Wie hat star. We actually did that in the previous problem. This why Hat Star is equal to the white hat star in this situation. Um, so we got 5046.67 This is equal to 50 for 6.67 And now we need to come up with the tea at Alfa over two. So are Alfa is going to be one minus 10.99 because we're looking at a 99% confidence interval tickle 0.1 So this is equal to, um is there a 0.5? So, uh, we will come back to this part later. Let's first calculate the standard deviation for the independence. So the standard deviation of an independent, um, why value is equal to the regular standard deviation times the square root of one plus one over end plus, um, our given X value. So ex star next star minus our mean X square over the sum of the difference between each of our individual X values and our explore on. So first, let's calculate the mean of our exes. So the mean of our exes is going to be equal to 400 plus 4 50 plus 5 50 plus 600 plus 707 50 divided by six. And we get an X bar of 575. So this over here and this is 5 75 and this is 5 75 and we know that our ex star is equal to 500. So 500 and we know that we have six, um, elements in our data set. Our end is six. We know what each of the individual values and our data set are. All we need to do is come up with a standard deviation. The standard deviation is equal to the square root of the mean squares due to air this on a different page. Mean square due to error on the mean swear due to error is equal to the sum of squares due to error over the number of elements minus two and the sum of squares due to air is equal to the sum of each individual. Why? Value minus, um, the predicted value at that point squared. So Ah, we already discovered the predicted Why values over here on just that by plugging in all these X values into that X. So, um, we just have to take the difference between, um, he's given these actual why values and our predicted estimates. So this is going to be equal to, um, our first example. 4000 minus 4286.67 plus 5000 minus 4600 66. 666.67 and so on and so forth until you get to the end of the data set and then we get a sum of squares due to error. He going 233,333 0.33 So now that this is the sum of squares did error, we can plug that in. Here it is 233333.33 That's a lot of threes, and we have six elements in our data sets of six minutes to is equal to four. So we get mean, squared error of 58,333 0.33 And now, to find our standard deviation, just take the square root of that on the square root of 58,333 0.33 is equal to 241 for the 241 pulling 5 to 29 So now we can plug this into our, um, our estimate here. So I'm going to delete that. And we just got a value of two for one 0.5 to 2 nights. So this is equal to, um, I'll just read it out in cleaner terms. Times the square root of one plus 1/6 plus 500 minus 5 75 squared over. And now, on a different page, let's calculate this value some of each of the individual X values minus the mean of X squared. So this would be since our exports 5 25 4 100 minus 5 75 squared. Plus, the next value is for 54 50 minus 5 75 squared and so on and so forth until we get to the end of our data set, and then we get a, um Aaaah Difference. A sum of square differences equaling 93,000 750. So here, 93,750. Finally, we come up with a, um, standard deviation of thea individual value equaling. Um, where is it? Ah, 267 point five. Approximately 267.5. Um, now we just have to plug that into this formula after we find our t value. So t of Alfa over two. And our outfit this is equal to t 0.5 And, um, we have to find a key value associated with this. Ah, And since we have two degrees of freedom and our degrees of freedom or a ridiculous rumors n minus two, since we have an end of six, this is equal to four. So with four degrees of freedom and teach statistic at 0.5 Alfa over to that is equal to 4.60 for So now we can find a 99% ah production interval, which is equal to our test statistic of our our point estimate of 5046.67 plus or minus for point 604 times. Um, are the the standard deviation of the individual data point of 267.5 and then we get a interval of 381,000. 3815.1062 Sorry. Two. 66,278 0.2338 So this is our 99% prediction interval for the total cost for a next one. And now we're asked in port de or apart Caesar in part C to come up with, um Aaaah or to figure out whether the manager should be concerned about incurring such a high total cost for the month. So they find out that the actual total cost um, there y value is equal to 6000. Should they be concerned? And ah, no, they should not be concerned. No, they should not be concerned because hour interval contains the value 12,000 goes in here because we are 99% confident that the true um, prediction of next month's no production costs is between 303,815 and 6278 and 6000 lies between those two values. Managers should not be

View More Answers From This Book

Find Another Textbook

16:11

In a manufacturing process the assembly line speed (feet per minute) was tho…

05:57

Refer to exercise $5,$ where the following data were used to imvestigate whe…

01:59

In exercise $8,$ data on $x=$ temperature rating $\left(\mathrm{F}^{\circ}\r…

03:38

The data in the following table show the number of shares selling (millions)…

05:59

The following data are the monthly salaries $y$ and the grade point averages…

12:53

In exercise 18 the data on grade point average and monthly salary were as fo…

02:12

In exercise $1,$ the following estimated regression equation based on 10 obs…

05:45

Out-of-state tuition and fees at the top graduate schools of business can be…

09:38

The U.S. Department of Energy's Fuel Economy Guide provides fuel effici…

06:26

The results of Computerworld's Annual Job Satisfaction Survey showed th…