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The Apex corporation produces corrugated paper. It has collected monthly data from January 2001 through March...

The Apex corporation produces corrugated paper. It has collected monthly data from January 2001 through March 2003 on the following two variables:

y= total manufacturing cost per month (In thousands of dollars) (COST)

x= total machine hours used per month (Machine)

The data are shown below.

y x

1102 218
1008 199
1227 249
1395 277
1710 363
1881 399
1924 411
1246 248
1255 259
1314 266
1557 334
1887 401
1204 238
1211 246
1287 259
1451 286
1828 389
1903 404
1997 430
1363 271
1421 286
1543 317
1774 376
1929 415
1317 260
1302 255
1388 281

Answer the following question

Fill in the blanks for the following statement: “I am 95% confident that the average manufacturing cost at the Apex corporation for all months with 350 total machine hours is between ____ and ____.”

please show me steps

Solutions

Expert Solution

This problem can be solved using regression.

Following are the steps

1 calculate regression coefficient using mentioned formula
2 calculate yhat using regression formula
3 calculate root mean square error(syx)
4 calculate standard error for prediction
5 calculate t critical
6 calculate lower and upper value of interval

Step 1.

y = a+bx

To calculate a and b

y x xy x^2 y^2
1102 218 240236 47524 1214404
1008 199 200592 39601 1016064
1227 249 305523 62001 1505529
1395 277 386415 76729 1946025
1710 363 620730 131769 2924100
1881 399 750519 159201 3538161
1924 411 790764 168921 3701776
1246 248 309008 61504 1552516
1255 259 325045 67081 1575025
1314 266 349524 70756 1726596
1557 334 520038 111556 2424249
1887 401 756687 160801 3560769
1204 238 286552 56644 1449616
1211 246 297906 60516 1466521
1287 259 333333 67081 1656369
1451 286 414986 81796 2105401
1828 389 711092 151321 3341584
1903 404 768812 163216 3621409
1997 430 858710 184900 3988009
1363 271 369373 73441 1857769
1421 286 406406 81796 2019241
1543 317 489131 100489 2380849
1774 376 667024 141376 3147076
1929 415 800535 172225 3721041
1317 260 342420 67600 1734489
1302 255 332010 65025 1695204
1388 281 390028 78961 1926544

sum(y) =40424,sum(x^2) = 2703831, sum(x) = 8337, sum(x*y) = 13023399

a = 206.8651,b = 4.178798

regression equation : y = 206.8651+4.178798*x

Step2

Calculate yhat for x0= 350

= 206.8651+4.178798*350 = 1669.445

Step3

Calculate RMSE

residual = yhat - y where yhat is calculated by putting value of x in above formula

yhat residual residual^2
1117.843 -15.8432 251.0057
1038.446 -30.446 926.9583
1247.386 -20.3859 415.5852
1364.392 30.60774 936.8337
1723.769 -13.7689 189.5831
1874.206 6.794342 46.16309
1924.351 -0.35124 0.123368
1243.207 2.792892 7.800245
1289.174 -34.1739 1167.855
1318.425 -4.42548 19.58486
1602.584 -45.5838 2077.88
1882.563 4.436745 19.68471
1201.419 2.580875 6.660917
1234.85 -23.8495 568.7992
1289.174 -2.17389 4.725797
1402.001 48.99855 2400.858
1832.418 -4.41767 19.51585
1895.1 7.90035 62.41554
2003.748 -6.74841 45.54099
1339.319 23.68053 560.7675
1402.001 18.99855 360.9451
1531.544 11.45581 131.2355
1778.093 -4.0933 16.75507
1941.066 -12.0664 145.5988
1293.353 23.64731 559.1954
1272.459 29.5413 872.6886
1381.107 6.892547 47.5072

Syx = 21.78281

Step4:

calculate standard error for prediction

= 4.227521

here n= 27

step5:

calculate t critical = t table at 0.05 and degrees of freedom = n-2 = 27-2=25

t critical = 2.059539

Step 6:

calculate confidence interval

lower = 1660.738

upper = 1678.151

therefore

Fill in the blanks for the following statement: “I am 95% confident that the average manufacturing cost at the Apex corporation for all months with 350 total machine hours is between 1660.738 and 1678.151.”

milcah answered 1 year ago
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