Question

A sample of 200 voters over the age of 60 were asked whether they thought Social Security benefits should be increased for people over the age of 65. A total of 95 of them answered yes. A sample of 150 voters aged 18–25 were asked the same question and 63 of them answered yes. A pollster wants to know whether the proportion of voters who support an increase in Social Security benefits is greater among older voters. He will use the ᵯ = 0.05 level of significance.

Answer 1

Out of 200 voters over the age of 60 , 95 of them answered 'yes' ( n1 = 200 , x1 = 95)

Sample proportion = = x₁/n₁ = 95/200 = 0.475

Out of 150 voters aged 18–25, 63 of them answered 'yes' ( n2 = 150 , x2 = 63 )

Sample proportion = = x₂/n₂ = 63/150 = 0.42

Hypothesis :

Let , P₁ be the population proportion of voters  over the age of 60 who answered ' Yes"

P₂ be the population proportion of voters aged 18–25 who answered ' Yes"

We have to test whether the proportion of voters who support an increase in Social Security benefits is greater among older voters. i.t P1 > P2

( claim )

Right tailed test.  

Test statistic :

Where , p is pooled proportion .

So, test statistic is ,

Critical value :

Given , level of significance = = 0.05

Critical value for this right tailed test is ,

= = 1.645 { Using Excel function , =NORMSINV(1-0.05)=1.645 }

Rejection region = { z : z > 1.645}

Decision about null hypothesis :

It is observed that test statistic ( z = 1.0232) is less than critical value ( = 1.645 )

So, fail to reject null hypothesis.

Conclusion :

There is not enough evidence to conclude that proportion of voters who support an increase in Social Security benefits is greater among older voters.