A researcher conducted an experiment to see if specific eye exercises can improv
ID: 3229188 • Letter: A
Question
A researcher conducted an experiment to see if specific eye exercises can improve peripheral vision. A random sample of 5 people were rated for peripheral vision on a scale from 1 to 20, where 9 is considered average and higher scores indicate better peripheral vision. Then they followed the prescribed eye exercise program and were rated again. The results follow below.
Subject
1
2
3
4
5
Before
9
8
7
10
6
After
10
9
11
12
9
Test the hypothesis that the exercises improved peripheral vision. Use a 5% significance level.
a. List the null and alternate hypotheses in terms of the appropriate population parameter using =, <, or >.
b. Find the critical value(s) which determine(s) the rejection and acceptance regions.
c. Compute the relevant test statistic and the P-value.
d. Draw a conclusion. Do you reject the null hypothesis or not?
Subject
1
2
3
4
5
Before
9
8
7
10
6
After
10
9
11
12
9
Explanation / Answer
To test for equal population variance:
Ho = There is no difference between the two population variance i.e., B2 = 2A
H1 = There is difference between the two population variance i.e., B2 2A
F test = SB2 / SA2 = 2.5/1.7 = 1.4706
F/2(4,4) = 6.388234
Since Ftab = 6.388234 > Fcal =1.4706, there is no evidence to reject null hypothesis. Therefore, there is no difference between the two population variance.
To test for equal population mean:
Ho = There is no difference between the two population mean i.e., µB = µA
H1 = There is difference between the two population mean i.e., µB µA
XB = 8.0 ; nB = 5 ; sB2 = 2.5
XA = 10.2 ; nA = 5 ; sA2 = 1.7
Critical value= t,(nB+nA-2) = t0.05,(5+5-2) = 1.860 ( from t table)
Since the population variances are equal, the test statistic is given by:
t = (XB - XA) / [Sp (1/nB + 1/nA)] where Sp is the pooled standard deviation
Sp2 = (nB -1) sB2 + (nA -1) sA2 / [nB+nA-2] =4*2.5 +4*1.7/8 = 2.1
Sp = 2.1 =1.4491
t= (8.0 – 10.2 )/[1.4491 0.4] = -2.40046
p value = 0.02158
Since, tcal = -2.4004 < ttab = -1.860 and p-value =0.02158 < 0.05, there is no evidence to accept the null hypothesis. Therefore, we can conclude that the exercises has improved the peripheral vision.
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