Question

A study of the properties of metal plate-connected trusses used for roof support† yielded observations on axial-stiffness index (kips/in.) for plate lengths 4, 6, 8, 10, and 12 in. The axial stiffness data produced the following ANOVA output from Minitab.
Analysis of Variance for stiffness
Source DF SS MS F P
Length 4 43993 10998 10.48 0.000
Error 30 31475 1049
Total 34 75468
Level N Mean StDev
4 7 333.21 36.59
6 7 368.06 28.57
8 7 375.13 20.83
10 7 407.36 44.51
12 7 437.17 26.00
Pooled StDev = 32.39
Tukey's pairwise comparisons
Family error rate = 0.0500
Individual error rate = 0.00693
Critical value = 4.10
Intervals for (column level mean) - (row level mean)
4 6 8 10
6 −85.0
15.4
8 −92.1 −57.3
8.3 43.1
10 −124.3 −89.5 −82.4
−23.9 10.9 18.0
12 −154.2 −119.3 −112.2 −80.0
−53.8 −18.9 −11.8 20.4
(a) Is it plausible that the variances of the five axial stiffness index distributions are identical? Explain.
Yes, it is plausible because the median standard deviation (s2 = 28.57) is within 20 of the largest (s4 = 44.51) and smallest (s3 = 20.83) standard deviation.
Yes, it is plausible because the largest standard deviation (s4 = 44.51) is slightly more than twice the smallest (s3 = 20.83).
No, it is not plausible because the largest standard deviation (s4 = 44.51) is slightly more than twice the smallest (s3 = 20.83).
No, it is not plausible because the median standard deviation (s2 = 28.57) is not within 10 of the largest (s4 = 44.51) and smallest (s3 = 20.83) standard deviation.
(b) Use the output (without reference to our F table) to test the relevant hypotheses.
Fail to reject H0. There is no difference in axial stiffness for the different plate lengths.
Fail to reject H0. There is a difference in axial stiffness for the different plate lengths.
Reject H0. There is a difference in axial stiffness for the different plate lengths.
Reject H0. There is no difference in axial stiffness for the different plate lengths.
(c) Use the Tukey intervals given in the output to determine which means differ, and construct the corresponding underscoring pattern. (Select all that apply.)
4 and 6
4 and 8
4 and 10
4 and 12
6 and 8
6 and 10
6 and 12
8 and 10
8 and 12
10 and 12
There are no significant differences.

Answer 1

Answer and Step-by-step explanation:

a. Yes, since the highest standard deviation (s4 = 44.51) is significantly more than double the lowest (s3 = 20.83) is plausible.

hence, the second option is correct

(b)  The testing of relevant hypothesis using the output is shown below:

As we see that

The 0.05 is more than the F test P-value i.e 0.0000

So ultimately P-value is less

Therefore we have to reject the null hypothesis

Since p-value is less than 0.05 so we reject the null hypothesis.

Therefore, the third option is correct

(c)  The construction is shown below:

Since the sample size is the same so for Tuley's HSD it would also same i.e for each pair

Now Tuley's HSD is

$HSD=4.10\cdot \sqrt{\frac{1049}{7}}=50.19$

The table that reflects the difference in means in different level is as follows

Levels       Difference in mean

4-6             34.85

4-8             41.92

4-10            74.15

4-12           103.96

6-8            7.07

6-10           39.3

6-12          69.11

8-10           32.23

8-12           62.04

10-12            29.81

Now the table of significant difference is

Levels       Difference in mean

4-10                 74.15

4-12                103.96

6-12                 69.11

8-12                 62.04