In a random sample of 1982 residential properties purchased

In a random sample of 1,982 residential properties purchased this year, 198 were purchased as vacation homes. Suppose that 5 years ago, 11% of all residential properties were purchased as vacation homes. Do the data suggest that things have changed since 5 years ago? Use a = 0.05.

Solution

Set Up Hypothesis
Null, H0:P=0.11
Alternate, H1: P!=0.11
Test Statistic
No. Of Success chances Observed (x)=198
Number of objects in a sample provided(n)=1982
No. Of Success Rate ( P )= x/n = 0.0999
Success Probability ( Po )=0.11
Failure Probability ( Qo) = 0.89
we use Test Statistic (Z) for Single Proportion = P-Po/Sqrt(PoQo/n)
Zo=0.0999-0.11/(Sqrt(0.0979)/1982)
Zo =-1.4372
| Zo | =1.4372
Critical Value
The Value of |Z | at LOS 0.05% is 1.96
We got |Zo| =1.437 & | 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 != -1.43721 ) = 0.15066
Hence Value of P0.05 < 0.1507,Here We Do not Reject Ho

[ANSWERS]
Test statistic : Zo =-1.44
P-Value = 0.15066