A study is conducted to determine whether there is a significant difference in u

Question

A study is conducted to determine whether there is a significant difference in union membership based on sex of the person. Random samples of 5000 factory-employed men were polled, and of this group, 785 were members of a union. A random of 3000 factory-employed women were also polled, and of this group, 327 were members of a union.

a) Use the p-value at a level of significance of 0.05 to test the hypothesis that the there is a significant difference in union membership based on sex of the person.

b) Construct the confidence interval that is relevant to this problem. Use the same level of significance, and draw conclusion (explain) about the hypothesis using the C.I.

c) If p1 = .15 and p2 = 0.12, the analyst wishes to have a probability of Type II error of 0.10, and same type I error as in parts (a) and (b). How many union members of each sex should be surveyed?

Explanation / Answer

A.
Test For Significance of Difference Of Proportion
Null Hypothesis, There Is No Significance between them Ho: p1 = p2
Alternate Hypothesis, There Is Significance between them H1: p1 != p2
Test Statistic
Sample 1 : X1 =785, n1 =5000, P1= X1/n1=0.157
Sample 2 : X2 =327, n2 =3000, P2= X2/n2=0.109
Finding a P^ value For Proportion P^=(X1 + X2 ) / (n1+n2)
P^=0.139
Q^ Value For Proportion= 1-P^=0.861
we use Test Statistic (Z) = (P1-P2)/(P^Q^(1/n1+1/n2))
Zo =(0.157-0.109)/Sqrt((0.139*0.861(1/5000+1/3000))
Zo =6.008
| Zo | =6.008
Critical Value
The Value of |Z | at LOS 0.05% is 1.96
We got |Zo| =6.008 & | 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 != 6.008 ) = 0
Hence Value of P0.05 > 0,Here we Reject Ho

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 )=785
No.Of Observed (n1)=5000
P1= X1/n1=0.157
Proportion 2
No. of chances(X2)=327
No.Of Observed (n2)=3000
P2= X2/n2=0.109
C.I = (0.157-0.109) ±Z a/2 * Sqrt( (0.157*0.843/5000) + (0.109*0.891/3000) )
=(0.157-0.109) ± 1.96* Sqrt(0)
=0.048-0.015,0.048+0.015
=[0.033,0.063]
ANS:there is a significant difference in union membership based on sex of the person.

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