Question

Researchers in a populous country contacted more than​ 25,000 inhabitants aged 25 years to see if they had finished high​ school; 88.5 % of the 12, 499 males and 80.7​% of the 12, 846 females indicated that they had high school diplomas.

​a) What assumptions are necessary to satisfy the conditions necessary for​ inference?

​b) Create a 99​% confidence interval for the difference in graduation rates between males and​ females, p Subscript males Baseline minus p Subscript females.

​c) Interpret your confidence interval.

​d) Is there evidence that boys are more likely than girls to complete high​ school?

Answer 1

Researchers in a populous country contacted more than​ 25,000 inhabitants aged 25 years to see if they had finished high​ school;

x1: Number of males indicated that they had high school diplomas

x2 = Number of females indicated that they had high school diplomas

p1=proportion of males indicated that they had high school diplomas

p2 = proportion of females indicated that they had high school diplomas

88.5 % of the 12, 499 males and 80.7​% of the 12, 846 females indicated that they had high school diplomas.

n1 = 12499 ; p1 = 0.885 , n2 = 12846 , p2 = 0.807

a)

Condition for the confidence interval for proportion:

1) sample is random

2) for x1 : n1p1 > 15 & n1(1-p1) > 15

n1p1 = 12499 * 0.885 = 11062 > 15

n1(1-p1) = 12499 * 0.115= 1437 > 15

for x2 :- n2p2 > 15 & n2(1-p2)

n2p2 = 12846*0.807 = 10367 > 15

n2(1-p2) = 12846 * 0.193 =2479 > 15

Therefore condition satisfied.

b) 99% confidence interval for the difference between two proportion:

alpha = 0.01

Zc = Zalpha/2 = Z 0.01/2 = Z0.005 = 2.576

C)

99 % confidence interval for p1-p2 is ( 0.066,0.09) which indicates that we are 99% confident that the true difference between population proportions is contained by the interval (0.066, 0.09)

d)

We want to test that proportion of boys who complete school is more than that of girls

H0: P1 - P2 = 0

Vs

H1: P1 -P2 > 0

Our confidence interval is (0.066,0.09)

So we fail to reject Ho if the confidence interval contains null hypothesis value.(here 0)

Here confidence interval does not contain 0.

So we will reject Ho.

We may conclude that the data provides sufficient evidence to conclude that the proportion of boys who complete school is more than that of girls.

i.e. there is evidence that boys are more likely than girls to complete high​ school.