Answer to Question #153774 in Statistics and Probability for vishnu

Question

In a sample of 500 persons from Town A, 200 are found to be consumers of Wheat. In a sample of 400 from Town B, 220 are found to be consumers of Wheat. Do these data reveal a significant difference between Town A and Town B as far as the proportion of Wheat consumers is concerned?

Answer 1

H₀: Two towns do not differ much as far as the proportion of wheat consumption. P₁ = P₂

H1: P₁ ≠ P₂

The Statistic test is

$Z = \frac{p_1-p_2}{\sqrt{PQ(\frac{1}{n_1}+ \frac{1}{n_2})}} \\ p_1 = \frac{200}{500} =0.4 \\ p_2 = \frac{220}{400} = 0.55 \\ P = \frac{n_1p_1+n_2p_2}{n_1+n_2} \\ = \frac{500(0.4) + 400(0.55)}{500+400} \\ = \frac{200+220}{900} \\ = 0.466 \\ Q = 1 -P = 1 -0.466 = 0.534 \\ Z = \frac{0.4-0.55}{\sqrt{(0.466)(0.534)(\frac{1}{500}+ \frac{1}{400})}} \\ = \frac{0.15}{0.03346} \\ = 4.48$

Calculated value Z = 4.48

Tabulated value of Z at 5% level of significance for a two tail test is 1.96

Calculated value >Tabulated value, H₀ is rejected.

Hence the data reveal a significant difference between town A and town B so far as the proportion of wheat consumers is concerned.