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TEST OF HYPOTHESES FOR EQUALITY OF PROPORTIONS OF TWO POPULATIONS
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 z0 = .
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.
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 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
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.
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