Question

4. Does distance from object affect the eye focus time? An industrial engineer is conducting an experiment on eye focus time. She is interested in the effect of the distance of the object from the eye on the focus time. Four distances (4’, 6’, 8’, 10’) will be studied. She has 5 subjects available for the experiment, and has decided that she will test each subject at each of the 4 distances; the order in which the distances are tested will be randomly decided for each study participant. The focus time measurements are given in the table below and are posted on in a text file Eyedata.

2

(a) What type of experimental design was used in this experiment?

(b) What statistical model would you use to analyze these data?

(c) What assumptions would be required for the model in (b)?

(d) What evidence do these data provide for / against the hypothesis that the distance has no effect on focus time?

(e) What contrast would you use to test for a difference between the mean focus times for distances of 4’ and 6’? Find a 95% confidence bound for this contrast and interpret.

(f) Test if there is a significant difference between the 4’ and 6’ group with the 8’and 10’ group. Clearly state the contrast you will use, find its unbiased estimator and standard error of the estimator. Next perform the test.

Subject Distance FocusTime

1 4 10

2 4 6

3 4 6

4 4 6

5 4 6

1 6 7

2 6 6

3 6 6

4 6 1

5 6 6

1 8 5

2 8 3

3 8 3

4 8 2

5 8 5

1 10 6

2 10 4

3 10 4

4 10 2

5 10 3

Answer 1

a). The experimental design used a single factor randomized block design .In this design ,five subjects (called as block) were used and treated with with a single condition change of distance variation by keeping them at 4 different distances (4',6,',8',10') . The bloacks were observed for focal distance at random and hence we term it as randomized single factor bloack design.

We will use the single factor ANOVA to ascertain if there is a significant change in focal distance amongst the blocks .

So we will now build the hypothesis testing .

We will assume that there is no difference between the means of the focal distances observed by 5 blocks from 4 different positions.

Let the means be

We conduct ANOVA test based on the above model and get the following results

Anova: Two-Factor Without Replication
SUMMARY	Count	Sum	Average	Variance
1	4	28	7	4.666667
2	4	19	4.75	2.25
3	4	19	4.75	2.25
4	4	11	2.75	4.916667
5	4	20	5	2
4	5	34	6.8	3.2
6	5	26	5.2	5.7
8	5	18	3.6	1.8
10	5	19	3.8	2.2
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Rows	36.3	4	9.075	7.117647	0.003548	3.259167
Columns	32.95	3	10.98333	8.614379	0.002543	3.490295
Error	15.3	12	1.275
Total	84.55	19

From the ANOVA we can see that

1) the p value is <0.05 and hence we can say with more than 95 % confidence that there is no variation in focal distance with change in block and distance .

e) for checking the difference between means for 4' and 6' we will use the t test for difference in mean between two independant samples.

H0: u1 - u2 = 0, where u1 is the mean of first population and u2 the mean of the second.

Ha : u1 - u2 is not equal to 0