About 30 of all burglaries are through an open or unlocked d

About 30% of all burglaries are through an open or unlocked door or window. A random sample of 130 burglaries indicated that 85 were not via an open or unlocked door or window. At the 0.05 level of significance, can it be concluded that this differs from the stated proportion?

Solution

Set Up Hypothesis
Null, H0:P=0.3
Alternate, H1: P!=0.3
Test Statistic
No. Of Success chances Observed (x)=85
Number of objects in a sample provided(n)=130
No. Of Success Rate ( P )= x/n = 0.6538
Success Probability ( Po )=0.3
Failure Probability ( Qo) = 0.7
we use Test Statistic (Z) for Single Proportion = P-Po/Sqrt(PoQo/n)
Zo=0.65385-0.3/(Sqrt(0.21)/130)
Zo =8.8039
| Zo | =8.8039
Critical Value
The Value of |Z | at LOS 0.05% is 1.96
We got |Zo| =8.804 & | 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 != 8.80393 ) = 0
Hence Value of P0.05 > 0,Here we Reject Ho

It can be concluded that this differs from the stated proportion