Question

We would like to test H_0: μ x ≥ μ y in a paired setting. The sample data shows the sample mean of X is larger than the sample mean of Y. And the sample covariance satisfies s_xy >0. Comparing the unpaired two-sample test vs the paired two-sample test, which test shall yield a smaller p-value?

The paired test

Same p-value

Cannot be determined

The unpaired test

Answer 1

Sol:

took

X	Y
12	5
20	6
24	7
35	8

use data >data analysis>regression

you will get

t-Test: Paired Two Sample for Means
	X	Y
Mean	22.75	6.5
Variance	91.58333	1.666666667
Observations	4	4
Pearson Correlation	0.98478
Hypothesized Mean Difference	0
df	3
t Stat	3.914905
P(T<=t) one-tail	0.014813
t Critical one-tail	2.353363
P(T<=t) two-tail	0.029625
t Critical two-tail	3.182446

p=0.014813(for paired)

t-Test: Two-Sample Assuming Unequal Variances
	X	Y
Mean	22.75	6.5
Variance	91.58333	1.666667
Observations	4	4
Hypothesized Mean Difference	0
df	3
t Stat	3.365572
P(T<=t) one-tail	0.021777
t Critical one-tail	2.353363
P(T<=t) two-tail	0.043553
t Critical two-tail	3.182446

p=0.021777(for independent t test)

From this example

The paired test yields less p value of 0.0148 compared to 0.021777 for The unpaired test (p value is 0.021777)

ANSWER:

The paired test