Question

Researchers were curious on how accurate people are at judging distances in different body orientations. Participants viewed a target shown in front of them while "standing" (upright), "supine" (lying down on their back), or "prone" (lying on their stomach). They had to judge how far away the target was from them (in centimetres), then the researchers recorded the errorsthey made.

Using the data below, conduct a repeated measure ANOVA to determine whether different body orientations make any difference in people's judgment of distance. Use α = .05. Report η², and if the ANOVA is significant conduct a post-hoc test using Tukey’s HSD. Finally, please state all the steps and the formulas you used and report your conclusion clearly in words.

When submitting, please put your answer in the field provided and upload images of your work. Up to 3 images may be uploaded. I strongly encourage you to do your work in tables to help structure your work and keep it easily visible in your images.

* unreadable or blurry images will not be graded

** If there are any discrepancies between the answer entered in the field and the answer on the images, only answer on the images will be graded

Subject          Standing          Supine          Prone

1                        -1                     -4                 -2

2                        -3                     -2                 -3

3                         2                       2                 -1

4                         0                     -2                 -3

5                         2                     -1                   0

6                        -3                     -5                 -4

7                         4                       1                  0

8                        -2                     -3                 -2

9                         0                     -1                 -3

10                      -1                     -4                 -3

∑X²                    48                     81                61

SS                     47.6                  44.9             16.9

Answer 1

column data , i (Factor B)	standing	supine	prone
count, ni =	10	10	10
mean , x̅ i =Σxi / ni	-0.200	-1.900	-2.100
std. dev., si =	2.30	2.23	1.37
sample variances, si^2 =	5.289	4.989	1.878
SS ,Σ(x-Xbar)²	47.6	44.9	16.9
grand mean , x̅̅ =	ΣXij/Σn =
square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	1.4400	0.2500	0.4900
SS(between)= SSB = Σni( x̅ i - x̅̅)² =	14.4000	2.5000	4.9000

		Factor A
count, n j	mean , x̅ j =Σxj/nj	std. dev., sj	sample variances, sj^2	( x̅ j - x̅̅)²	nj( x̅j - x̅̅)²
3	-2.33	1.5275	2.333	0.871	2.613
3	-2.67	0.5774	0.333	1.604	4.813
3	1.00	1.7321	3.000	5.760	17.280
3	-1.67	1.5275	2.333	0.071	0.213
3	0.3	1.5275	2.333	3.004	9.013
3	-4.0	1.0000	1.000	6.760	20.280
3	1.6666667	2.0817	4.333	9.404	28.213
3	-2.333333	0.5774	0.333	0.871	2.613
3	-1.333333	1.5275	2.333	0.004	0.013
3	-2.666667	1.5275	2.333	1.604	4.813

	SSA = total=	89.867
	SSA=Σnj( x̅j - x̅̅)² =	89.867
SST =	Σ(Xij - X̅̅ ) =	131.2
SSE =	SST-SSA-SSB =	19.533
ΣXi² =	190
SST=Σxi² - (Σxi)²/N=		131.2

so, SSA=	89.867
SSB=	21.8
SSE=	19.533
df factor A(row) = r-1 =	9
df factor B(column) = c-1 =	2
here,total observations, N =	30
df error = (r-1)(c-1) =	18
mean square factor A , MSA = SSA/df=	9.99
mean square(factor B) =MSB= SSB/df =	10.90
mean square error = MSE =SSE/df =	1.09

ANOVA
Source of Variation	SS	df	MS	F-stat	p-value		F-critical	Result
Rows	89.8667	9	9.9852	9.201	0.0000		2.46	significant
Columns	21.8000	2	10.9000	10.044	0.0012		3.555	significant
Error	19.5333	18	1.0852
Total	131.2000	29
						α =	0.05

/.......................

tukey test:

Level of significance	0.05
no of treatments	3
df error	18
MSE	1.0852
q-statistic value	3.6093

			confidence interval
population mean difference	critical value	lower limit	upper limit	result
µ1-µ2	1.7	1.19	0.51	2.89	means are different
µ1-µ3	1.9	1.19	0.71	3.09	means are different
µ2-µ3	0.2	1.19	-0.99	1.39	means are not different

....................

Please let me know in case of any doubt.

Thanks in advance!

Please upvote!