Question

An engineer wants to know if the mean strengths of three different concrete mix designs differ significantly. He also suspects that slump may be a predictor of concrete strength. Slump is a measure of the uniform ity of the concrete, with a higher slump indicating a less uniform mixture.The accompanying data represent the 28-day strength (in pounds per square inch) of three different mixtures with three different slumps. Complete parts (a) through Click the icon to view the data. Mixture 67-0-400 Slump Mixture 67-0-301 Mixture 67-0-353 3,960 4,815 4,595 4.495 3.75 4,005 4,145 (a) Is the requirement of equal population variances provided? 3,445 4,185 4,585 Yes 4,010 4.070 3,855 4.545 3415 3,675 No 3,710 4,175 4.010 (b) Determine whether there is significant interaction between mixture type and slump. Choose the correct answer belo 3.290 4.020 3,875 5 3.390 4.355 3,700 O There is no significant interaction. O There is significant interaction. 3,740 3,935 3,350 (c) If there is not significant interaction, determine whether there is significant difference in the means for the three types of mixture. Choose the correct answer below O A. There is significant difference in the means. O B. There is no significant difference in the means O C. There is significant interaction If there is not significant interaction, determine whether there is significant difference in the means for the slumps. Choase the correct answer below O A. There is significant O B. interaction. There is significant difference in the means. (d) Draw an interaction plot of the data to support the results of parts (b) and (c). Choose the correct graph below. O A. B. C. AMean AMean AMean 67-0-301 67-0-301 4500- 4500- 4500- 67-0-400 67-0-400 67-0-400 4000- 4000- 4000- 67-0-353 67-0-301 67-0-353 67-0-353 3500- 3500- 3500- Slump Slump Slump (e) If there is significant difference in the means for the three mixture types, use Tukey's test to determine which pairwise means differ using a familywise error rate of a0.05. Choose the correct answer below OA. The Tukey's test indicates that the mean 28-day strength of mixture 67-0-353 is significantly different from the mean 28-day strength of the other two mixtures O B. The Tukey's test indicates that the mean 28-day strength of mixture 67-0-400 is significantly different from the mean 28-day strength of the mixture 67-0-301 O C. There is no significant difference in the means for the three mixture types. O D. The Tukey's test indicates that the mean 28-dav strength of mixture 67-0-400 is

Answer 1

a) test for equal variances

Test for Equal Variances: Strength versus Mixture

Method

Null hypothesis	All variances are equal
Alternative hypothesis	At least one variance is different
Significance level	α = 0.05

Test for Equal Variances: Strength versus slump

Method	Test Statistic	P-Value
Multiple comparisons	—	0.777
Levene	0.35	0.708

Test for Equal Variances: Strength vs slump

Method	Test Statistic	P-Value
Multiple comparisons	—	0.541
Levene	0.38	0.688

Test for Equal Variances: Strength vs Mixture

requirement of equal variances is provided.

b)

ANOVA test

General Linear Model: Strength versus slump, Mixture

Method

Factor coding

(-1, 0, +1)

Factor Information

Factor	Type	Levels	Values
slump	Fixed	3	1, 2, 3
Mixture	Fixed	3	1, 2, 3

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
slump	2	1179702	589851	8.60	0.002
Mixture	2	1760946	880473	12.83	0.000
slump*Mixture	4	267304	66826	0.97	0.446
Error	18	1235100	68617
Total	26	4443052

There is no significant interaction because p -value is more than 0.05.

c) Yes there is a significant difference in three types of mixture.

Yes there is a significant difference in three types of slump.

d)

Interaction Plot for Strength Data Means Mixture 4500 2 4250 4000 3750 3500 3 1 2 slump Mean

e)

Tukey Simultaneous Tests for Differences of Means

Difference of Mixture Levels	Difference of Means	SE of Difference	Simultaneous 95% CI	T-Value	Adjusted P-Value
2 - 1	626	123	(310, 941)	5.07	0.000
3 - 1	314	123	(-1, 629)	2.54	0.051
3 - 2	-312	123	(-627, 4)	-2.52	0.053

Individual confidence level = 98.00%

results indicate that option b is correct.

mixture 67-0-400 is significantly differ from mixture 67-0-301.