STAT-3001-5/STAT-3001L-SISTAT-3001P-6-Statistical Methods2017 Winter Qtr 11/27-0
ID: 3350335 • Letter: S
Question
STAT-3001-5/STAT-3001L-SISTAT-3001P-6-Statistical Methods2017 Winter Qtr 11/27-0 Test: Week 6 Exam This Question: 2 pts 2/18-PT3 Time Remaining: 04 40 49 Submit Test 28 of This Test: 100 pts possib The values in the table below are measured occurred with dick the icon to view the table. Find the P-value (in millimeters) of male skulls from different epochs Changes in head shape over time suggest thet mmigrant populations. Use a 0 05 significance level to test the claim that the different epochs all have the same mean O A. There is sufficient evidence to reject the claim that the different epochs have the same mean O B. There is not sufficient evidence to reject the claim that the different epochs have the same mean 4000 B.C 1850 B.C 150 AD. 128 138 136 130 138 125 129 133 135 132 134 137 129 134 136 137 137 127 136 138 134 141 142 137 145 Click to select your answer(s)Explanation / Answer
The problem is analysed by one-way ANOVA with equal number of observations per block.
p-value = 0.03 ANSWER 1
Conclusion: option A ANSWER 2
Elaborate details are presented below for better understanding and comprehension of the working of the solution.
Back-up Theory
Suppose we have data of a 1-way classification ANOVA, with r rows/treatments and n observations per cell/treatment.
Let xij represent the jth observation in the ith row, j = 1,2,…,n; i = 1,2,……,r
Then the ANOVA model is: xij = µ + i + ij, where µ = common effect, i = effect of ith row, and ij is the error component which is assumed to be Normally Distributed with mean 0 and variance 2.
Null hypothesis: H0: 1 = 2 = 3 = 0 Vs Alternative: HA: H0 is false
Now, to work out the solution,
Terminology:
Row total = xi.= sum over j of xij
Grand total = G = sum over i of xi.
Correction Factor = C = G2/N, where N = total number of observations = r x n =
Total Sum of Squares: SST = (sum over i,j of xij2) – C
Row Sum of Squares: SSR = {(sum over i of xi.2)/(n)} – C
Error Sum of Squares: SSE = SST – SSR
Mean Sum of Squares = Sum of squares/Degrees of Freedom
Fcrit: upper % point of F-Distribution with degrees of freedom n1 and n2, where n1 is the DF for Row and n2 is the DF for Error
Significance: Fobs is significant if Fobs > Fcrit
ANOVA TABLE
Source of
Variation
Degrees of Freedom (DF)
Sum of squares (SS)
Mean Sum
of squares
(MS = SS/DF)
Fobs
Fcrit*
Significance**
Row
r - 1
SSR
MSR/MSE
Error
rn - r
SSE
Total
rn - 1
SST
NOTE:
* Fcrit: upper % point of F-Distribution with degrees of freedom n1 and n2, where n1
is the DF for Row and n2 is the DF for Error
** Significance: Fobs is significant if Fobs > Fcrit
Calculations
Let xij = maximum breadth of male skull belonging to epoch i, i = 1 for BC 2000, 2 for BC 1850 and 3 for AD 150 and j = 1, 2, ….. , 9.
Excel calculation summary:
alpha
0.05
#treat
3
n1
9
n2
9
n3
9
n
27
x1.
1193
x2.
1208
x3.
1243
G = x..
3644
C
491805
Sx1j^2
158273
Sx2j^2
162256
Sx3j^2
171853
Sxij^2
492382
Sxi.^2/ni
491951.3
SST
576.963
SSR
146.2963
SSE
430.6667
ANOVA TABLE
Source
DF
SS
MS
F
Fcrit
p-value
Epoch
2
146.3
73.14815
4.076367
3.4028261
0.0299171
Error
24
430.67
17.94444
Total
26
576.96
22.19088
Since p-value < 0.05 (given level of significance), H0 is rejected. => there is not enough evidence to suggest that the mean skull breadths remained the same over the epochs.
DONE
Source of
Variation
Degrees of Freedom (DF)
Sum of squares (SS)
Mean Sum
of squares
(MS = SS/DF)
Fobs
Fcrit*
Significance**
Row
r - 1
SSR
MSR/MSE
Error
rn - r
SSE
Total
rn - 1
SST
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