Question

(20) 9. It is believed that 20% or more of the student population of ABC College is over age 25. Suppose 98 students were randomly surveyed to see if less than 20% of the population are over age 25.

A. What is the null hypothesis using symbols? What is the null hypothesis using words?

B. What is the alternate hypothesis using symbols? What is the alternate hypothesis using words?

C. What type of error happens (if any) if the researcher stated that more than 20% of the student population of ABC College is over age 25, but in fact less than 20% is over age 25? Explain.

D. What type of error happens (if any) if the researcher stated that less than 20% of the student population of ABC College is over age 25, but in fact exactly 20% is over age 25? Explain.

E. Suppose the probability of a type I error is 11% and the probability of a type II error is 8%. What is the power of the test?

F. Is a type I error or a type II error more critical if it is important to know that we do not have more than 20% of the population that is over age 25?

Answer 1

Solution :

A) Null hypothesis using symbol :

10 :p 0.20

Null hypothesis using words :

H_{​​​​​​0} : The proportion of student population of ABC college who are over age 25 is greater than or equal to 0.20.

B) Alternative hypothesis using symbol :

$\large H_{1}: p < 0.20$

Alternative hypothesis using words :

H_{​​​​​1} : The proportion of student population of ABC college who are over age 25 is less than 0.20.

C) The error committed by not rejecting the null hypothesis, when actually the null hypothesis is false is known as type II error.

If the researcher stated that more than 20% of the student population of ABC College is over age 25, but in fact less than 20% is over age 25, it means he is not rejecting the null hypothesis when actually the null hypothesis is false. Hence, the researcher is making type II error.

d) The error committed by rejecting the null hypothesis, when actually the null hypothesis is true is known as type I error.

If the researcher stated that less than 20% of the student population of ABC College is over age 25, but in fact exactly 20% is over age 25, it means the researcher is rejecting the null hypothesis when actually the null hypothesis is true. Hence, the researcher is making type I error.

E) Power of test = 1 - P(Type II error)

We have, P(Type II error) = 8% = 0.08

Hence,

Power = (1 - 0.08) = 0.92

Power of the test is 0.92.