Question

A study of the properties of metal plate-connected trusses used for roof support yielded the following observations on axial stiffness index (kips/in.) for plate lengths 4, 6, 8, 10, and 12 in:

4:	340.2	409.5	311.0	326.5	316.8	349.8	309.7
6:	432.1	347.2	361.0	404.5	331.0	348.9	381.7
8:	390.4	366.2	351.0	357.1	409.9	367.3	382.0
10:	359.7	452.9	461.4	433.1	410.6	384.2	362.6
12:	413.4	441.8	419.9	410.7	473.4	441.2	465.8

Does variation in plate length have any effect on true average axial stiffness? State the relevant hypotheses using analysis of variance.

H₀: μ₁ ≠ μ₂ ≠ μ₃ ≠ μ₄ ≠ μ₅
H_a: at least two μ_i's are equal H₀: μ₁ ≠ μ₂ ≠ μ₃ ≠ μ₄ ≠ μ₅
H_a: all five μ_i's are equal     H₀: μ₁ = μ₂ = μ₃ = μ₄ = μ₅
H_a: all five μ_i's are unequal H₀: μ₁ = μ₂ = μ₃ = μ₄ = μ₅
H_a: at least two μ_i's are unequal

Test the relevant hypotheses using analysis of variance with α = 0.01. Display your results in an ANOVA table. (Round your answers to two decimal places.)

Source	Degrees of freedom	Sum of Squares	Mean Squares	f
Treatments	2	3	4	5
Error	6	7	8
Total	9	10

Give the test statistic. (Round your answer to two decimal places.)
f = 11

What can be said about the P-value for the test?

P-value > 0.100 0.050 < P-value < 0.100     0.010 < P-value < 0.050 0.001 < P-value < 0.010 P-value < 0.001

State the conclusion in the problem context.

Reject H₀. There are no differences in the true average axial stiffness for the different plate lengths. Reject H₀. There are differences in the true average axial stiffness for the different plate lengths.     Fail to reject H₀. There are differences in the true average axial stiffness for the different plate lengths. Fail to reject H₀. There are no differences in the true average axial stiffness for the different plate lengths.

Answer 1

(1)
Correct option:

H0: All five 's are equal:

Ha: At least two 's are unequal:

(2)

From the given data, the following Table is calculated:

	4	6	8	10	12	Total
N	7	7	7	7	7	35
	2363.5	2606.4	2623.9	2864.5	3066.2	13524.5
Mean	2363.5/7=337.6429	2606.4/7=372.3429	2623.9/7=374.8429	2864.5/7=409.2143	3066.2/7=438.0286	13524.5/35=386.414
	805385.91	978186.6	986087.31	1182648.83	1346811.94	5299120.59
Std. Dev.	35.0405	35.852	20.5635	41.7423	24.929	46.3556

From the above Table, ANOVA Table is calculated as follows:

Source	Degrees of freedom	Sum of Squares	Mean squares	F
Treatments	4	41261.0086	41261.008/4=10315.2541	10315.2541/1059.9858=9.7315
Error	30	31799.5743	31799.5743/30=1059.9858
Total	34	73060.5829

Test statistic F = 10315.2541/1059.9858=9.73

(3)

Degrees of Freedom (4,30)

By Technology, p - value = 0.000036

So

Correct option:

P - Value < 0.001

(4)

Correct option:

Reject H0. There are differences in the true average axial stiffness for the different plate lengths.