Hypothesis Testing Of The Difference Between Proportions

Illustration

Ken industrial producer have manufacture a perfume termed as "fianchetto." In order to test its popularity or reputation in the market, the manufacturer carried a random surveyor study in back rank city whereas 10,000 consumers were interviewed after that 7,200 demonstrated preferences. The manufacturer moved also to area Rook town where he interviewed 12,000 consumers out of that 1,0000 demonstrated preference for the product.

Required

Design a statistical test and thus use it to advise the manufacturer regarding the differences in the proportion, at 5 percent level of significance.

Solution

H₀ : π₁ = π₂

H₁ : π₁ ≠ π₂

The critical value for this two tailed test at 5 percent level of significance = 1.96.

Now Z = ¦{(P₁ - P₂) - (Π₁ - Π₂)/S(P₁ - P₂)}¦

But as the null hypothesis is π₁ = π₂, the second part of the numerator disappear that is

π₁ - π₂ = 0 which will usually be the case at this level.

Then Z = ¦ {(P₁ - P₂)/S (P₁ - P₂)}¦

Whereas:

Here S (P₁ - P₂) = √{(pq/n₁) + √(pq/n₁)}   

Whereas p = (p₁n₁ + p₂n₂)/ (n₁ + n₂)

And q = 1 - p;

∴ in our case

P = {10,000 (0.72) + 12,000 (0.83)}/(10,000 + 12,000)

= 84,000/22,000

= 0.78

∴ q = 0.22

S (P₁ - P₂) = √{(0.78(0.22)/10,000) + (0.78(12,000)/12,000)}

= 0.00894

= ¦(0.72 - 0.83)/0.00894¦  

=12.3

As 12.3 > 1.96, we reject the null hypothesis however accept the alternative. The differences among the proportions are statistically significant. It implies that the perfume is more popular in Rook town rather than in Back rank city.