Question

Cycle III of the Health Examination Survey used a nationwide probability sample of youths age 12 to 17. One object of the survey was to estimate the percentage of youths who were illiterate. A test was developed to measure literacy. It consisted of seven brief passages, with three questions about each, like the following:

There were footsteps and a knock at the door. Everyone inside stood up quickly. The only sound was that of the pot boiling on the stove. There was another knock. No one moved. The footsteps on the other side of the door could be heard moving away.

• The people inside the room

(a) Hid behind the stove

(b) Stood up quickly

(c) Ran to the door

(d) Laughed out loud

(e) Began to cry

• What was the only sound in the room?

(a) People talking

(b) Birds singing

(c) A pot boiling

(d) A dog barking

(e) A man shouting

• The person who knocked at the door finally

(a) Walked into the room

(b) Sat down outside the door

(c) Shouted for help

(d) Walked away

(e) Broke down the door

This test was designed to be at the fourth-grade level of reading, and subjects were defined to be literate if they could answer more than half the questions correctly. There turned out to be some difference between the performance of males and females on this test: 7% of the males were illiterate, compared to 3% of the females. Is this difference real, or the result of chance variation? You may assume that the investigators took a simple random sample of 1,600 male youths, and an independent simple random sample of 1,600 female youths.

To answer, provide the following information (round to 3 decimals if necessary):

(1) The SE for the percentage of the males who were illiterate in the male sample:

(2) The SE for the percentage of the females who were illiterate in the female sample:

(3) the SE for the difference

(4) Z-value:

(5) Is it possible to explain this difference as a chance variation? Answer yes or no:

Answer 1

sample #1   ----->  
first sample size,     n1=   1600
proportion success of sample 1 , p̂1=   0.0700
      
sample #2   ----->  
second sample size,     n2 =    1600
proportion success of sample 1 , p̂ 2=   0.0300

SE = √(p^*(1-p^)/n)

1) SE for males =SQRT(0.07*0.93/1600) = 0.006378675

2) SE for females=SQRT(0.03*0.97/1600) = 0.004264681

3) SE for difference= √(SEmales + SE females) = 0.0077

4) Z-statistic = (p̂1 - p̂2)/SE = (   0.040 -0 ) /   0.0077   =   5.19

5)

p-value =        0.0000
decision :    p-value<α,Reject null hypothesis   

ANSWER : NO