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# Test of Hypotheses for Equality of Proportions of two Populations

General Procedure for Test of Hypotheses for Equality of Proportions of two Populations: Procedure Steps, Example Solved Problems

TEST OF HYPOTHESES FOR EQUALITY OF PROPORTIONS OF TWO POPULATIONS

## Procedure:

Step 1 : Let PX and PY denote respectively the proportions of Population-1 and Population-2 possessing the qualitative characteristic (attribute) under study. Frame the null hypothesis as H0: PX=PY and choose the suitable alternative hypothesis from

(i) H1: PX≠ PY (ii) H1: PX>PY (iii) H1: PX<PY

Step 2 : Let p X and pY denote respectively the proportions of the samples of sizes m and n drawn from Population-1 and Population-2 possessing the attribute, where m and n are large (i.e., m ≥ 30 and n ≥ 30). Also, mpX > 5, m (1- pX) > 5, npY > 5 and n (1 - pY ) > 5 .

Here, these two samples are assumed to be independent.

Step 3 : Specify the level of significance, α.

Step 4 : Consider the test statistic  under H0. Here, . The approximate sampling distribution of the test statistic under H0  is the N(0,1) distribution.

Step 5 : Calculate the value of Z for the given data as z.

Step 6 : Choose the critical value, ze, corresponding to α and H1 from the following table

Step 7 : Decide on H0 choosing the suitable rejection rule from the following table corresponding to H1.

## Example 1.13

A study was conducted to investigate the interest of people living in cities towards self-employment. Among randomly selected 500 persons from City-1, 400 persons were found to be self -employed. From City -2, 800 persons were selected randomly and among them 600 persons are self-employed. Do the data indicate that the two cities are significantly different with respect to prevalence of self-employment among the persons? Choose the level of significance as α = 0.05.

## Solution:

Step1 : Let PX and PY be respectively the proportions of self-employed people in City-1 and City-2. Then, the null and alternative hypotheses are

Null hypothesis: H0 : PX = PY

i.e., there is no significant difference between the proportions of self-employed people in City-1 and City-2.

Alternative hypothesis: H1 : PX PY

i.e., difference between the proportions of self-employed people in City-1 and City-2 is significant. It is a two-sided alternative hypothesis.

Step 2 : Data

The given sample information are

Here, m 30, n 30, mpX = 400 > 5, m(1− pX) = 100 > 5, npY = 600 > 5 and n(1− pY) = 200 > 5.

Step 3 : Level of significance

α= 5%

Step 4 : Test statistic

The test statistic under the null hypothesis is

The sampling distribution of Z under H0 is the N(0,1) distribution.

Step 5 : Calculation of Test Statistic

The value of Z for given sample information is calculated from

z0 = 2.0764

Step 6 : Critical value

Since H1 is a two -sided alternative hypothesis, the critical value at 5% level of significance is ze = 1.96.

Step 7 : Decision

Since H0 is a two-sided alternative, elements of the critical region are determined by the rejection rule |z0| > ze. Thus, it is a two-tailed test. For the given sample information, ze = 2.0764 > ze = 1.96. Hence, H0 is rejected. We can conclude that the difference between the proportions of self-employed people in City-1 and City-2 is significant.

Tags : Procedure Steps, Example Solved Problems | Statistics , 12th Statistics : Chapter 1 : Tests of Significance - Basic Concepts and Large Sample Tests
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12th Statistics : Chapter 1 : Tests of Significance - Basic Concepts and Large Sample Tests : Test of Hypotheses for Equality of Proportions of two Populations | Procedure Steps, Example Solved Problems | Statistics