Tests based on paired samples

Suppose that a random sample of n matched pairs of observations is obtained from populations with means and . The observations will be denoted by .

Let

and

denote the observed sample mean and standard deviation for the n differences . Let us denote difference between two population means by . In this case test statistic will be calculated as

If the population differences is a normal distribution, then the following tests have significance level

1. To test either null hypothesis

or

against the alternative

the decision rule is

Reject if

2. To test either null hypothesis

or

against the alternative

the decision rule is

Reject if

3. To test the null hypothesis

against the two sided alternative

the decision rule is

Reject if or

Here, is the number for which

where the random variable follows a Student’s t distribution with degrees of freedom.

Remark: When we want to test the null hypothesis that the two population means are equal, we set .

Example:

A medical researcher wishes to determine if a pill has the undesirable side effect of reducing the blood pressure of the user. The study involves recording the initial blood pressures of 7 college age adults. After they use the pill regularly for three month, their blood pressures are again recorded. The researcher wishes to draw inferences about the effect of the pill on blood pressure from the information given in table

Before
After

Do the data substantiate the claim that use of the pill reduces the blood pressure? Use . Assume that the population of paired differences has a normal distribution.

Solution:

Let d be the difference between the pressures before and after using pills.

d=before –after=

The necessary calculations are shown in the following table

Before	After	Difference
		-3

The values of and are calculated as follows:

.

Let be the mean blood pressure for all adults before and -after using the pill.

The null and alternative hypotheses are

(no difference)

against

(mean decreases)

The decision rule is that

Reject if

Since 2.792<3.14, we accept and make conclusion at the level 0.01 that using pills does not affect blood pressure.

Дата добавления: 2015-08-05