Question

A researcher wants to determine a model to predict the 28-day strength of a concrete mixture. The following data represent the slump (a measure of concrete uniformity in inches) x1, 7-day strength (in pounds per square inch, or psi) x2, and the 28-day strength (in psi) y of a certain type of concrete. Slump measure of the uniformity of the concrete, with a higher slump indicating a less uniform mixture. Complete parts (a) through (j). 3.75 4.5 2.5 4 3 2.5 4.5 4,25 х 4 3080 1770 2050 3360 2250 2240 2640 3810 3350 38754025 46853890 3765 4985 4475 3610 4535 2590 2220 2510 2330 Х Y ) Construct 95% confidence and prediction intervals for concrete for which slump is 3.0 inches and 7-day strength is 2600 psi. Interpret the results. Find the 95% confidence interval for the sample. lower bound 4272.6 upper bound 4604.8 (Round to one decimal place as needed.) Find the 95% prediction interval for the sample. lower bound 3887.7 upper bound 4989.7 (Round to one decimal place as needed.) Based on these results, it can be said with 95% confidence that the 28-day strength for a specific sample of concrete for which slump is 3.0 inches and 7-day strength is 2600 psi will be between about 3741.2 and 5136.2 psi and the mean 28-day strength of all concrete for which slump is 3.0 inches and 7-day strength is 2600 psi will be between about 4155.9 and 4721.6 psi.

Answer 1

Output using excel:

Regression Statistics
Multiple R	0.8721
R Square	0.7605
Adjusted R Square	0.7006
Standard Error	276.5048
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	2	1942115.4562	971057.7281	12.7011	0.0033
Residual	8	611639.0892	76454.8862
Total	10	2553754.5455
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3890.5497	828.7504	4.6945	0.0016	1979.4478	5801.6516
x1	-295.9255	109.8519	-2.6939	0.0273	-549.2445	-42.6065
x2	0.5523	0.2171	2.5441	0.0345	0.0517	1.0529

formula used:

A 1 confidence Interval Estimate and Pre Data 4 Confidence Level l0.95 1 x1 given value x2 given value 3 2600 SE(CIEPIDatal$As2:sc$121.CIEPIDatal$A$2:SC$12) =MMULT( TRANSPOSE talsA$2:$c$12).CIEPIDatalSA$2:$c$12) -MMULT ( TRANSPOSE(CI =MMULT I TRANSPOSE(CIEPIDatal$A$2:SC$12), CIEPIData!$A$2: SC$12)- MMULT(TRANSPOSE(CIEPIDatal$A$2:Sc$12), CIEPIData!$A$2:SC$12 )-MMULT(TRANSPOSE(CIEPIData!$A$2: SC$12),CIEPIData!SA$2: $c$12) =MMULT (TRANSPOSE(CIEPIData!$A$2:SC$12),CIEPIData!$A$2: SC$12 )- MMULT(TRANSPOSE(CIEPIData!$A$2: $c$12),CIEPIData!$A$2:SC$12)- MMULT(TRANSPOSE(CIEPIData!$A$2:SC$12),CIEPIData!SA$2: $c$12) IDatalSA$2:SC$12) -MMU 10 11 12 = MINVERSE ( SBS9:$0$11) 13 Inverse of X'X MINVERSE (SB59:S0S11) MINVERSE($B$9:$D$11) S$9-SDS11) $9 $D$11) =MINVERSE89 MVERSE(SaS9-SD$11) 15 $11) 16 17 x'G times Inverse of x'x MMULT(TRANSPOSE(SBS5:S8$7),SBS13:SD$15) -MMULT(TRANSPOSE[$B$5 :$B$71 , $B$13:$D$15) MMULT(TRANSPOSE(SB$5:SB$7 ), $8$13:$D$15) 18 =MMULT( SB$ 17 :$D$17,S8$5: $8$ 7) 2T (1- B4, COMPUTEISB$13) 19 [x'G times Inverse of XX] times XG 21 d dY (YHat) $7), COMPUTE!$B$17:$B$19) 22 23 For Average Predicted Y (YHat) 820 SQRT( B19) COMPUTEISB$7 -R21 -B24 24 Interval Half Width 25 Confidence Interval Lower Limit -R21 + B24 Confidence Interval Upper Limit 27 28 For Individual Response Y |-B20 SQRT(1+B19)*COMPUTE!I$B$7 -B21 - B29 -821 + 829 29 Interval Half Width 30 Prediction Interval Lower Limit 31 Prediction Interval Upper Limit

A D 1 Confidence Interval Estimate and Prediction Interval 2 Data 4 Confidence Level 95% 5 1 6 x1 given value 7 x2 given value 2600 8 9 x'X 11 43 27040 10 43 176.125 103832.5 11 2704 103832.5 68526600 12 13 Inverse of X'X 8.983432 -0.969531 -0.002076 14 -0.969531 0.157837 0.000143 15 0.000143 0.000001 -0.002076 16 17 X'G times Inverse of X'X 0.677924 -0.123150 -0.000043 18 19 [x'G times Inverse of X'X] times XG 0.19677112 20 tStatistic 2.30600414 21 Predicted Y (YHat) 4438.71069 22 23 For Average Predicted Y (YHat) 24 Interval Half Width 282.8417 25 Confidence Interval Lower Limit 4155.87 26 Confidence Interval Upper Limit 27 4721.55 28 For Individual Response Y 29 Interval Half Width 697.5386 30 Prediction Interval Lower Limit 31 Prediction Interval Upper Limit 3741.17 5136.25 32

95% confidence interval :

Lower Bound = 4155.9

Upper bound = 4721.6

95% prediction interval :

Lower Bound = 3741.2

Upper bound = 5136.2