In: Statistics and Probability
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.
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.