Suppose that a particular production process is stable if th

Suppose that a particular production process is stable, if there are at most 2% defective items. Let p be the true proportion of defective items.

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Suppose that a particular production process is stable, if there are at most 2% defective items. Let p be the true proportion of defective items. (a) We sample n = 200 items at random and consider hypotheses testing about p. Formulate null and alternative hypotheses. (b) What is your conclusion of the above test, if one observes 3 defective items out of 200? Note: you have to choose appropriate a.

Solution

a)
Set Up Hypothesis
Null, there are greater 2% defective items H0:P>=0.02
Alternate, there are at most 2% defective items H1: P<0.02
b)
Test Statistic
No. Of Success chances Observed (x)=3
Number of objects in a sample provided(n)=200
No. Of Success Rate ( P )= x/n = 0.015
Success Probability ( Po )=0.02
Failure Probability ( Qo) = 0.98
we use Test Statistic (Z) for Single Proportion = P-Po/Sqrt(PoQo/n)
Zo=0.015-0.02/(Sqrt(0.0196)/200)
Zo =-0.5051
| Zo | =0.5051
Critical Value
The Value of |Z | at LOS 0.05% is 1.64
We got |Zo| =0.505 & | Z | =1.64
Make Decision
Hence Value of |Zo | < | Z | and Here we Do not Reject Ho
P-Value: Left Tail -Ha : ( P < -0.50508 ) = 0.30675
Hence Value of P0.05 < 0.69325,Here We Do not Reject Ho

we don\'t have evidence to indicate that there are only 2% defective items