Many polls have asked people whether they are trying to lose

Many polls have asked people whether they are trying to lose weight. A Gallup Poll in November of 2008 showed that 22% of men said they were seriously trying to lose weight- In 2006, 24% of men (with the same average weight of 194 pounds as the men polled in 2008) said they were seriously trying to lose weight. Assume that both samples contained 500 men. Determine how many men in the sample from 2008 and how many in the sample from 2006 said they were seriously trying to lose weight. Determine whether the difference in proportions is significant al the 0.05 Ievel. Repeat the problem with the same proportions but assuming both sample sizes are now 5000. Comment on the different p-values and conclusions with different sample sizes.

Solution

a.
Sample 1 : X1 =110, n1 =500, P1= X1/n1=0.22
Sample 2 : X2 =120, n2 =500, P2= X2/n2=0.24
b.
Null Hypothesis, There Is No Significance between them Ho: p1 = p2
Alternate Hypothesis, There Is Significance between them H1: p1 != p2
Test Statistic
Finding a P^ value For Proportion P^=(X1 + X2 ) / (n1+n2)
P^=0.23
Q^ Value For Proportion= 1-P^=0.77
we use Test Statistic (Z) = (P1-P2)/(P^Q^(1/n1+1/n2))
Zo =(0.22-0.24)/Sqrt((0.23*0.77(1/500+1/500))
Zo =-0.751
| Zo | =0.751
Critical Value
The Value of |Z | at LOS 0.05% is 1.96
We got |Zo| =0.751 & | Z | =1.96
Make Decision
Hence Value of |Zo | < | Z | and Here we Do not Reject Ho
P-Value: Two Tailed ( double the one tail ) -Ha : ( P != -0.7514 ) = 0.4524
Hence Value of P0.05 < 0.4524,Here We Do not Reject Ho

c.
Sample 1 : X1 =1100, n1 =5000, P1= X1/n1=0.22
Sample 2 : X2 =1200, n2 =5000, P2= X2/n2=0.24
Finding a P^ value For Proportion P^=(X1 + X2 ) / (n1+n2)
P^=0.23
Q^ Value For Proportion= 1-P^=0.77
we use Test Statistic (Z) = (P1-P2)/(P^Q^(1/n1+1/n2))
Zo =(0.22-0.24)/Sqrt((0.23*0.77(1/5000+1/5000))
Zo =-2.376
| Zo | =2.376
P-Value: Two Tailed ( double the one tail ) -Ha : ( P != -2.3762 ) = 0.0175
Hence Value of P0.05 > 0.0175,Here we Reject Ho

d.
With size 500, we Failed to Reject Ho
With size 5000, we Reject Ho