Question

A production line operation is tested for filling weight accuracy using the following hypotheses. Hypothesis Conclusion and Action H 0: = 16 Filling okay, keep running H a:       16 Filling off standard; stop and adjust machine The sample size is 39 and the population standard deviation is = 0.9. Use = .05. Do not round intermediate calculations.

a. What would a Type II error mean in this situation?

b. What is the probability of making a Type II error when the machine is overfilling by .5 ounces (to 4 decimals)?

c. What is the power of the statistical test when the machine is overfilling by .5 ounces (to 4 decimals)?

Answer 1

Answer :

given data :-

The sample size = 39

Standard deviation σ = 0.9

Use $\alpha$ = 0.05

the null and alternative hypothesis is

H0 = 16

Ha = 16

Here type II error mean that we fail to reject the null hypothesis the invalid speculation regardless of whether it false that implies we will neglect to recognize that in a generation line activity which is tried for filling weight precision, the genuine filling sum isn't 16 ounces.

now we need to find out the standard error of sample mean

we know that

standard error = σ/sqrt(n)

= 0.9/sqrt(39)

= 0.9/6.244

SE = 0.150

we will reject the null hypothesis if sample mean

x̅ = 16 + Z95% SE

x̅ = 16 + 1.96 * 0.150

x̅ = 16 + 0.282

x̅ = 16.294

Here true mean = 16.5 as machine is overfilling by 0.5 ounces

Pr(Type II error) = Pr(x̅ < 16.294 ; 16.5 ; 1/6)

Z = (16.294 - 16.5)/(0.15)

= -0.206/0.15

= -1.3733

Pr(Type II error) = Pr(x̅ < 16.294 ; 16.5 ; 0.15)

= Pr(Z < -1.3733)

= 0.0848

Power of the test = 1 - 0.0848

= 0.9152