The proportion of adults living in a small town who are college graduates is est
ID: 3220837 • Letter: T
Question
The proportion of adults living in a small town who are college graduates is estimated to be p = 0.6. To test this hypothesis, a random sample of 200 adults is selected. The fail to reject region is defined to be 110 lessthanorequalto x lessthanorequalto 130, where x is the number of college graduates in our sample. Test the null hypothesis that p = 0.6; against the alternative that p notequalto 0.6. Use the normal approximation. (a) Evaluate alpha assuming that p = 0.6. (b) Evaluate beta for the alternative p = 0.5. (c) What is the power of this test if the real p = 0.5? (d) Suppose that the sample of 200 adults contained 138 college graduates. What is the P-value?Explanation / Answer
A.a) Given,
H0:p=0.6
Ha:p=0.5
p=0.6
sample size(n)=200
to calculate (type 1 error)
P()=reject the null hypothesis if its true
mean(µ)=np=200*0.6=120
standard deviation(SD)=sqrt(npq)=sqrt(200*0.6*0.4)=6.9282
test statistic(Z) is given as
P(110<x<130)=P{(110-120)/6.9282<Z<(130-120)/6.9282}
P(110<x<130)=P(-1.4434<Z<1.4434)
P(110<x<130)=0.9251-0.0749=0.8502
type 1 error: alpha() is given as 1-0.8502=0.1498
A.b) Beta(type 2 error) is failing to reject Null if false
H0:p=0.6
Ha:p=0.5
p=0.6
sample size(n)=200
to calculate beta(type 2 error)
P(beta)= failing to reject Null if false and alternate is true
mean(µ)=np=200*0.5=100
standard deviation(SD)=sqrt(npq)=sqrt(200*0.5*0.5)=7.0711
test statistic(Z) is given as
P(110<x<130)=P{(110-100)/7.0711<Z<(130-100)/7.0711}
P(110<x<130)=P(-1.4142<Z<4.2426)
P(110<x<130)=1-0.9207 =0.0793
type 2 error: beta is given as 1-0.9207 =0.0793
A.c) power is calculated as
power=1-beta
power =1-0.0793=0.9207
power of the test is 0.9207
A.d) sample size=200
college graduates=138
p-value is
p=138/200=0.69
p-value is 0.69
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