Question

In an effort to test the hypothesis that the proportion of voters in the younger than 40 year old age bracket who will vote for a particular politician is different than the proportion voters in the above 40 age bracket, the following data was collected. Test hypothesis at 0.10 level of significance.

Population Number Who Will Vote for Politician Sample Size

Below 40 348 700
Above 40 290 650

Required:
a. Reject the Null Hypothesis and condlude that the proportion of voters in the younger than 40 year oid age bracket who will vote for a particular different than the proportion voters in the above 40 age bracket.
b. Do not reject the Null Hypothesis and conclude that the proportion of wotors in the younger than 40 year old age bracket who will vote for a particular politician is not different than the proportion voters in the above 40 age bracket.

Answer 1

We reject H₀ we support the claim that the two proportion are different

Explanation:

Younger than 40 years old

Sample size n₁ = 700

x₁ = 348

p₁ = 348 / 700 p₁ = 0,497 p₁ = 49,7 %

Above 40 years old

Sample size n₂ = 650

x₂ = 290

p₂ = 290/ 700 p₂ = 0,414 p₂ = 41,4 %

p = ( n₁*p₁ + n₂*p₂ ) / n₁ + n₂

p = 700 * 0,497 + 650 * 0,414 ) / 700 + 650

p = ( 347.9 + 269,1 ) / 1350

p = 0,457 p = 45,7 %

q = 1 - p q = 54,3 % q = 0,543

CI is 90 % significance level is 10 % α = 10 % α = 0,1 α/2 = 0,05

zscore for α/2 from z- table is : z(c) = 1,64

Test Hypothesis

Null Hypothesis H₀ p₁ = p₂

Alternative Hypothesis Hₐ p₁ ≠ p₂

To calculate z(s) = ( p₁ - p₂ ) / √ pq/n₁ + pq/ n₂

p₁ - p₂ = 0,497 - 0,414 = 0,083

√ pq/n₁ + pq/ n₂ = √ 0,457*0,543/ 700 + 0,457*0,543/ 650

√ pq/n₁ + pq/ n₂ = √ 3,545*10⁻⁴ + 3,82 * 10⁻⁴

√ pq/n₁ + pq/ n₂ = 2,71 * 10⁻² = 0,0271

z(s) = 0,083/ 0,0271

z(s) = 3,06

Comparing z(s) and z(c)

|z(s)| > |z(c)|

Then z(s) is in the rejection region and we reject H₀