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**

**Step 1 : **Let** ***P _{X}*

(i) *H _{1}: P_{X}≠ P_{Y}* (ii)

**Step 2 : Let ***p*** **_{X}** **and** ***p _{Y}*

Here, these two samples are assumed to be independent.

**Step 3 : **Specify the level of significance,** ***α*.

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

**Step 5 : **Calculate the value of *Z* for the given data as *z*_{0 }= .

**Step 6 : **Choose the critical value,** ***z _{e}*,
corresponding to

**Step 7 : **Decide on** ***H _{0}*

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.

**Step1 : **Let** ***P _{X}*

**Null hypothesis: ***H*_{0}** **:** ***P _{X}*

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

**Alternative hypothesis: ***H*_{1}** **:** ***P _{X}*

*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, *mp _{X}*

**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 *H*_{0}
is the *N(0,1)* distribution.

**Step 5 : ****Calculation of Test Statistic**

The value of *Z* for given sample information is calculated
from

z_{0}
= 2.0764

**Step 6 : ****Critical value**

Since *H _{1}* is a two -sided alternative hypothesis,
the critical value at 5% level of significance is

**Step 7 : ****Decision**

Since *H _{0}* is a two-sided alternative, elements of
the critical region are determined by the rejection rule |z

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

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