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8.15 Test for difference between proportions

If two samples are drawn from different populations, we may be interested in finding out whether the difference between the proportion of successes is significant or not. Let x1 and x2 be the number of items possessing the attribute A, in the random sampling of sizes n1 and n2 from two populations respectively.

Then the sample proportions of successes are   if P1 and P2 are proportion of successes in the two populations and Q1 = 1 - P1, Q2 = 1 - P2 then

Under the hypothesis that the proportions in two populations are equal.

i.e. P1 = P2 = P Þ Q1 = Q2 = Q (say) then

In general, however, we do not know the populationís proportion of success. In such a case we can replace P by its best estimate P = the pooled estimate of the actual proportion in the population, where

Pooled estimate (P) =

Example A machine produced 20 defective articles in a batch of 500. After overhauling it produced 3 defective in a batch of 100. Has the machine improved ?

Solution: Ho : P1 = P2 i.e. The machine has not improved after overhauling Ha : P1 Ļ P2

Now P1 = = 0.032 and P2 = = 0.030

Pooled estimate of actual proportion in the population is given by

Q = 1 - P = 0.968

Ho is true i.e. the machine has not improved significantly.


8.1 Population
8.2 Sample
8.3 Parameters and Statistic
8.4 Sampling Distribution
8.5 Sampling Error
8.6 Central Limit Theorem
8.7 Critical Region
8.8 Testing of Hypothesis
8.9 Errors in Tesitng of Hypothesis
8.10 Power o a Hypothesis Test
8.11 Sampling of Variables
8.12 Sampling of Attributes
8.13 Estimation
8.14 Testing the Difference Between Means
8.15 Test for Difference Between Proportions
8.16 Two Tailed and one Tailed Tests
8.17 Test of Significance for Small Samples
8.18 Students t-distribution
8.19 Distribution of 't' for Comparison of Two Samples Means Independent Samples
8.20 Testing Difference Between Mens of Two Samples Dependent Samples or Matched Paired Observations
8.21 Chi-Square
8.22 Sampling Theory of Correlation
8.23 Sampling Theory of Regression

Chapter 1

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