In a test of the reliability of products produced by two mac

In a test of the reliability of products produced by two machines, machine 1 produced 10 defective parts in a run of 200, while machine 2 produced 15 defective parts in a run of 260. Test the claim that proportion of defectives produced by machine 1 produces is less than that of machine 2. Use 5% level of significance. Suppose that the true value of the proportion of defectives produced by machine 1 and machine 2 are given by p1=0.04 and p2 = 0.08. Calculate the power for the test with level of significance, = 0.05. Also find the p-value. Please include some work so I can see how values are found.

Solution

Q1.
In a test of the reliability of products produced by two machines, machine 1 produced 10 defective parts in a run of 200, while machine 2 produced 15 defective parts in a run of 260. Test the claim that proportion of defectives produced by machine 1 produces is less than that of machine 2. Use 5% level of significance

Null,Ho: p1 = p2
Alternate, machine 1 produces is less than that of machine 2 H1: p1 < p2
Test Statistic
Sample 1 : X1 =10, n1 =200, P1= X1/n1=0.05
Sample 2 : X2 =15, n2 =260, P2= X2/n2=0.058
Finding a P^ value For Proportion P^=(X1 + X2 ) / (n1+n2)
P^=0.054
Q^ Value For Proportion= 1-P^=0.946
we use Test Statistic (Z) = (P1-P2)/(P^Q^(1/n1+1/n2))
Zo =(0.05-0.058)/Sqrt((0.054*0.946(1/200+1/260))
Zo =-0.361
| Zo | =0.361
Critical Value
The Value of |Z | at LOS 0.05% is 1.645
We got |Zo| =0.361 & | Z | =1.645
Make Decision
Hence Value of |Zo | < | Z | and Here we Do not Reject Ho
P-Value: Left Tail - Ha : ( P < -0.3608 ) = 0.35914
Hence Value of P0.05 < 0.35914,Here We Do not Reject Ho

We don\'t have evidence to support machine 1 produces is less than that of
machine 2

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