Two different simple random samples are drawn from two different populations. Th

Question

Two different simple random samples are drawn from two different populations. The first sample consists of 40 people with 22 having a common attribute. The second sample consists of 2000 people with 1433 of them having the same common attribute. Compare the results from a hypothesis test of p1=p2 (with a 0.05 significance level) and a 95% confidence interval estimate of p1p2.

1.Identify the test statistic

2.Identify the critical value(s).

3.What is the conclusion based on the hypothesis test?

4.The 95% confidence interval is [ ]<P1-P1<[ ]

Explanation / Answer

PART A.
Given that,
sample one, x1 =22, n1 =40, p1= x1/n1=0.55
sample two, x2 =1433, n2 =2000, p2= x2/n2=0.717
null, Ho: p1 = p2
alternate, H1: p1 != p2
level of significance, = 0.05
from standard normal table, two tailed z /2 =
since our test is two-tailed
reject Ho, if zo < -1.96 OR if zo > 1.96
we use test statistic (z) = (p1-p2)/(p^q^(1/n1+1/n2))
zo =(0.55-0.717)/sqrt((0.713*0.287(1/40+1/2000))
zo =-2.305
| zo | =2.305
critical value
the value of |z | at los 0.05% is 1.96
we got |zo| =2.305 & | z | =1.96
make decision
hence value of | zo | > | z | and here we reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != -2.3055 ) = 0.0211
hence value of p0.05 > 0.0211,here we reject Ho
ANSWERS
---------------
null, Ho: p1 = p2
alternate, H1: p1 != p2
test statistic: -2.305
critical value: -1.96 , 1.96
decision: reject Ho
p-value: 0.0211
PART B.
Confidence Interval for Diffrence of Proportion
CI = (p1 - p2) ± Z a/2 Sqrt(p1(1-p1)/n1 + p2(1-p2)/n2 )
Proportion 1
No. of chances( X1 )=22
No.Of Observed (n1)=40
P1= X1/n1=0.55
Proportion 2
No. of chances(X2)=1433
No.Of Observed (n2)=2000
P2= X2/n2=0.717
C.I = (0.55-0.717) ±Z a/2 * Sqrt( (0.55*0.45/40) + (0.717*0.284/2000) )
=(0.55-0.717) ± 1.96* Sqrt(0.006)
=-0.167-0.155,-0.167+0.155
=[-0.322,-0.011]

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