Test of Hypotheses for Population Proportion

TEST OF HYPOTHESES FOR POPULATION PROPORTION

Procedure:

Step 1 : Let P denote the proportion of the population possessing the qualitative characteristic (attribute) under study. If p₀ is an admissible value of P, then frame the null hypothesis as H₀:P = p₀ and choose the suitable alternative hypothesis from

(i) H₁: P ≠ p₀ (ii) H₁: P > p₀ (iii) H₁: P < p₀

Step 2 : Let p be proportion of the sample observations possessing the attribute, where n is large, np > 5 and n(1 – p) > 5.

Step 3 : Specify the level of significance, α.

Step 4 : Consider the test statistic Z   under H₀. Here, Q = 1 – P.

The approximate sampling distribution of the test statistic under H₀ is the N(0,1) distribution.

Step 5 : Calculate the value of Z under H₀ for the given data as 

Step 6 : Choose the critical value, z_e, corresponding to α and H₁ from the following table

Step 7 : Make decision on H₀ choosing the suitable rejection rule from the following table corresponding to H₁.

Example 1.12

A survey was conducted among the citizens of a city to study their preference towards consumption of tea and coffee. Among 1000 randomly selected persons, it is found that 560 are tea-drinkers and the remaining are coffee-drinkers. Can we conclude at 1% level of significance from this information that both tea and coffee are equally preferred among the citizens in the city?

Solution:

Step 1 : Let P denote the proportion of people in the city who preferred to consume tea.

Then, the null and the alternative hypotheses are

Null hypothesis: H ₀ : P = 0.5

i.e., it is significant that both tea and coffee are preferred equally in the city.

Alternative hypothesis: H ₁ : P ≠ 0.5

i.e., preference of tea and coffee are not significantly equal. It is a two-sided alternative hypothesis.

Step 2 : Data

The given sample information are

Sample size (n) = 1000. Hence, it is a large sample.

No. of tea-drinkers = 560

Sample proportion (p) = 560/1000 = 0.56

Step 3 : Level of significance

α= 1%

Step 4 : Test statistic

Since n is large, np = 560 > 5 and n(1 – p) = 440 > 5, the test statistic under the null hypothesis, is Z = .

Its sampling distribution under H₀ is the N(0,1) distribution.

Step 5 : Calculation of Test Statistic

The value of Z can be calculated for the sample information from

Thus, z₀ = 3.79

Step 6 : Critical value

Since H₁ is a two-sided alternative hypothesis, the critical value at 1% level of significance is z_α/2 = z_0.005 = 2.58.

Step 7 : Decision

Since H₁ is a two- sided alternative, elements of the critical region are determined by the rejection rule |z₀| ≥ z _e. Thus it is a two-tailed test. Since |z₀| = 3.79 > z_e = 2.58, reject H₀ at 1% level of significance. Therefore, there is significant evidence to conclude that the preference of tea and coffee are different.