Question

Listed below are brain volumes (cm3 ) of twins.

First Born	1005	1035	1281	1051	1034	1079	1104	1439	1029	1160
Second Born	963	1027	1272	1079	1070	1173	1067	1347	1100	1204

Test the claim at the 5% significance level that the brain volume for the first born is different from the second-born twin.

(a) State the null and alternative hypotheses.

(b) Find the critical value and the test statistic.

(c) Should H0 be rejected at the 5% significance level? Make a conclusion.

(d) Construct a 95% confidence interval for the paired difference of the population means

Answer 1

Solution:

We have to test the claim at the 5% significance level that the brain volume for the first born is different from the second-born twin.

Part a) State the null and alternative hypotheses.

Part b) Find the critical value and the test statistic.

n = number of pairs = 10

df= n - 1 = 10 - 1 = 9

Level of significance = 0.05

From t table for df = 9 and Two tail area = Level of significance = 0.05, t critical value = 2.262

Since this is two tailed test, t critical values are : -2.262 and 2.262

Test statistic:

where

Thus we need to make following table:

First Born	Second Born	di = Firs Born-Second Born	di^2
1005	963	42	1764
1035	1027	8	64
1281	1272	9	81
1051	1079	-28	784
1034	1070	-36	1296
1079	1173	-94	8836
1104	1067	37	1369
1439	1347	92	8464
1029	1100	-71	5041
1160	1204	-44	1936

Thus

Thus t test statistic:

Part c) Should H0 be rejected at the 5% significance level? Make a conclusion

No, since t test statistic value is neither less than -2.262 , nor greater than 2.262.

Thus we conclude that: there is not sufficient evidence to support the claim that: the brain volume for the first born is different from the second-born twin.

Part d) Construct a 95% confidence interval for the paired difference of the population means

Formula:

where

t_c is t critical value for c = 95% confidence level , that is for two tail area = 1 - c= 1 - 0.95 = 0.05 area.

df = n - 1 = 10 - 1 = 9

Thus t_c = 2.262

Thus

Thus

Thus a 95% confidence interval for the paired difference of the population means is inbetween :