In Design and Analysis of Experiments, 8th edition (John Wiley & Sons, 2012), D.
ID: 3055306 • Letter: I
Question
In Design and Analysis of Experiments, 8th edition (John Wiley & Sons, 2012), D. C. Montgomery described an experiment that determined the effect of four different types of tips in a hardness tester on the observed hardness of a metal alloy. Four specimens of the alloy were obtained, and each tip was tested once on each specimen, producing the following data:
a) Is there any difference in hardness measurements between the tips?
b) Use Fisher’s LSD method to investigate specific differences between the tips.
c) Analyze the residuals from this experiment.
Specimen e of Tip 93 93 9.3 9.4 9.2 9.7 9.4 9.3 9.4 9.6 9.6 9.8 9.5 10.0 10.0 9.9 9.7 10.2Explanation / Answer
Here is the MINITAB output for One-Way ANOVA(Unstacked):
Type of Tips 1 2 3 4 RESI1 RESI2 RESI3 RESI4 FITS1 FITS2 FITS3 FITS4
1 9.3 9.4 9.6 10.0 -0.1 -0.025 -0.125 0.05 9.4 9.425 9.725 9.95
2 9.4 9.3 9.8 9.9 0.0 -0.125 0.075 -0.05 9.4 9.425 9.725 9.
3 9.2 9.4 9.5 9.7 -0.2 -0.025 -0.225 -0.25 9.4 9.425 9.725 9.95
4 9.7 9.6 10.0 10.2 0.3 0.175 0.275 0.25 9.4 9.425 9.725 9.95
One-way ANOVA: 1, 2, 3, 4
Source DF SS MS F P
Factor 3 0.8250 0.2750 7.10 0.005
Error 12 0.4650 0.0387
Total 15 1.2900
S = 0.1969 R-Sq = 63.95% R-Sq(adj) = 54.94%
Individual 95% CIs For Mean Based on
Pooled StDev
Level N Mean StDev ---+---------+---------+---------+------
1 4 9.400 0.216 (--------*--------)
2 4 9.425 0.126 (--------*--------)
3 4 9.725 0.222 (--------*--------)
4 4 9.950 0.208 (--------*--------)
---+---------+---------+---------+------
9.25 9.50 9.75 10.00
Pooled StDev = 0.197
Grouping Information Using Fisher Method
N Mean Grouping
4 4 9.9500 A
3 4 9.7250 A B
2 4 9.4250 B C
1 4 9.4000 C
Means that do not share a letter are significantly different.
Fisher 95% Individual Confidence Intervals
All Pairwise Comparisons
Simultaneous confidence level = 81.57%
1 subtracted from:
Lower Center Upper -------+---------+---------+---------+--
2 -0.2783 0.0250 0.3283 (------*-----)
3 0.0217 0.3250 0.6283 (-----*------)
4 0.2467 0.5500 0.8533 (-----*-----)
-------+---------+---------+---------+--
-0.50 0.00 0.50 1.00
2 subtracted from:
Lower Center Upper -------+---------+---------+---------+--
3 -0.0033 0.3000 0.6033 (-----*-----)
4 0.2217 0.5250 0.8283 (-----*------)
-------+---------+---------+---------+--
-0.50 0.00 0.50 1.00
3 subtracted from:
Lower Center Upper -------+---------+---------+---------+--
4 -0.0783 0.2250 0.5283 (-----*------)
-------+---------+---------+---------+--
-0.50 0.00 0.50 1.00
This is not letting me paste the text in original format so this looks little scattered, but if you line up numbers it will all make sence.
Here p-value is 0.005 which is less than alpha(0.05), hence we reject the null hypothesis(there is no diffrence) and we can say that there is difference between tips.
Hope this helps.
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