# INTERVAL PREDICTION OF Y GIVEN A VALUE OF X Stats Assignment Homework Help

INTERVAL PREDICTION OF Y GIVEN A VALUE OF X

Population values Y, given a value of X, are values of a random variable they are not constants like the parameters A and B, or a value of JLl’ given  a value of X. We construct a prediction interval for values of a random variable. The formula for the standard error of predicting Y given X is  .

Thus we predict with 95 percent confidence that the number of gears made in 50 labor-hours will be between 30.1 and 43.5. Recall that in the previous example we estimated J-Ly more precisely; that is, we estimated with 95 percent confidence that the mean number of gears J-Ly made in 50 labor hours will be between 33.8 and 39.8.

EXERCISE See the last exercise. Construct a 90 percent prediction interval for the cost of processing an order for four items.

PROBLEMS

1. Colette Chicness, manager of the Cross Roads Hotel, collected sample data on x, the number of people who made room reservations before arriving at the hotel, and y, the number of these people who registered at the hotel. The data are given in Table A. The computed regression equation is 9 = -1.0093 + 0.8514x. (a) Compute the standard error of estimate. (b) Compute the standard error of the slope. (c) At the 1 percent level of significance, test the hypothesis that the slope ot the population regression line is zero .

2. Arne Nielsen, owner of Nielsen’s Farms. collected data on the number of bags of fertilizer x applied to a section of corn-seeded land, and the corn yield y, in bushels, from the section. All land sections were of the same size. The data are shown in Table B. The regression equation is 9 = 23.7333 + 5.8857x. (a) Compute the standard error of estimate. (b) Compute the standard error of the slope. (c) At the 10 percent level of significance, test the hypothesis that the increase in yield per additional  bag of fertilizer equals 5 bushels.

4. Joyce Laplace, an executive of Market Advice, Inc., wanted to deter . mine the relationship betweeny, the amount of money a family spends, and x, the amount of income the family receives. She expressed both x and y in thousands of dollars. For a sample of 12 families, Joyce computed the regression line 9 = 6.0785 + 0.6078x. She also computed l:x = 390, l:x2 = 13,400, and s, = 4.2243. (a) Perform a hypothesis test at the 1 percent level to determine whether spending and in comere related; that is, test the hypothesis that the slope of the population  regression line equals zero. (b) Construct the 95 percent confidence interval estimate of the slope of the population regression line. (c) Construct the 95 percent prediction interval estimate for the amount spent by a family having an income of \$40 thousand. (d) Construct the 95  percent confidence interval estimate for the mean amount spent by families whose income is \$40 thousand. (e) At the 5 percent level, perform a hypothesis test to determine whether the increase in the amount· spell from an additional \$1 thousand of income is greater than \$0.5 thousand. . 5. Pat Beaumont, president of Beauty Products Company, recorded 12 observations on adverting expenditure x and sales y. Pat recorded both expenditures and sales in thousands of dollars. Each observation was for a 3-month period. The regression equation is y = 40 + 2.8x. Also, LX = 96, LX2 = 854, and the standard error of estimate is 2.4. (a)
At the  5 percent level, test the hypothesis that the slope of the populationregression line is greater than 2. (b) What does the result of the test in (a) mean about the sales-advertising relationship? (c) Construct the
95 percent confidence interval estimate of the population slope. (d) Construct the 90 percent prediction interval estimate of sales for a 3-m nth period when advertising expenditure is 10 thousand dollars. (e) Construct the 90 percent confidence interval estimate of mean sales in 3-month periods when advertising expenditure is 10 thousand  dollars. 6. R. C. Cory, Inc., a large mail-order company, has a staff of  people who work on both inventory management and customer order filling. At the beginning of each working day it is necessasy to assign a certain number of these people to do order filling. For this purpose, a quick estimate of the number of orders in the early morning mail is needed. It was suggested that the estimate might be calculated from the weight of  was selected and,  for each day, a tabulation was made of the mail weight x, in hundreds of pounds, and the number of orders in the mail y, in thousands. Regre’ssioncalculation yielded LX = 210, LY = 530, LXY = 7225, LX2 = 2870, Ly2 = 18,190, a = 0.28948, b = 2.49624, and se = 0.263. (a) Construct the 95 percent prediction Interval estimate of the number of orders in the mail on a day when the weight of the mail is 15 hundred pounds. (b) Construct a 90 percent confidence interval estimate of the mean number of orders on days when the mail weighs 15 hundred pounds. (c) At the 5 percent level of significance, perform a test of the hypothesis that the slope of the regression line is less than 3. 7. In this problem, x is the Environmental Protection Agency (EPA) gas mileage (mpg) of a car in highway driving, and Y is mpg for the car in city driving. Calculations for 18 observations (cars) provided the following data: LX = 520, LY = 375, LXY = 11,250, LX2 = 15,680, and  Ly2’= 8150. The regression line isy = 2.5337 + 0.6334x. The standard error r of estimate is 2.14 mpg. (a) At the 1 percent level of significance,  test the hypothesis that the slope of the regression line isgreater than zero. (b) Construct a 99 percent confidence interval estimate
o f the slope of the population regression line. (c) Construct the 95pe~d,.; prediction interval estimate of the city mpg for a car getting 30 mpg in highway driving. (d) Construct the 95 percent confidence interval estimate of mean city mpg for cars which get 30 mpg in highway driving. 8. In thrs problem, X is city mpg and Y is combined city and highway mpg. For 18 cars, LX = 375, LY = 435, LXY = 9430, LX2 = 8150, Ly2 = 10,920, Y = 1.4812 + 1.0889x, and s, = 0.678 mpg. (a) At the 5 percent level of significance, test the hypothesis that the slope of the population regression line is less than 1. (Hint: Be careful when you decide .

Posted on August 27, 2014 in SIMPLE LINEAR REGRESSION AND CORRELATION
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