Question

Conduct a full hypothesis test and determine if the given claim is supported or not supported at the 0.05 significance level. A manufacturer considers his production process to be out of control when defects exceed 3%. In a random sample of 85 items, the defect rate is 5.8% but the manager claims that this is only a sample fluctuation and production is not really out of control. Test whether the manufacturer's claim that production is out of control is supported or not supported.

Answer 1

Solution:-

- The population proportion, P = 0.03 ( 3 % )

- The sample proportion, p = 0.058 ( 5.8 % )]

- The sample size , n = 85.

- State the hypothesis:

                   Null hypothesis : P = 0.03 ( 3 % )

                   Alternate hypothesis : P ≥ 0.03 ( 3 % )

- The rejection region is defined by the significance level α:

                         significance α = 0.05

                         Z-critical = Z_α = Z_0.05

                         p-value = Z_0.05 = 1.645

- The Z-statistics:

                   $Z = \frac{\sqrt{n}*(p - P ) }{\sqrt{P*(1-P)} }\\\\Z = \frac{\sqrt{85}*(0.058 - 0.03 ) }{\sqrt{0.03*(1-0.03)} }\\\\Z = 1.5132$

- Compare against the rejection value ( p - value ):

                   Z-critical = 1.645

                   Z-test = 1.5132

                   Z-test  < Z-critical  ... ( Null not rejected )

- We fail to reject the Null hypothesis, there is not enough evidence to support the manager's claim that the production process is out of control. Hence, we can support that the production process is not out of control.