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 33 and the population standard deviation is  = 0.7. Use  = .05. Do not round intermediate calculations.

1. What is the probability of making a Type II error when the machine is overfilling by .5 ounces (to 4 decimals)?
   
   
2. What is the power of the statistical test when the machine is overfilling by .5 ounces (to 4 decimals)?
   

Answer 1

Answer:

Given,

sample n = 33

standard deviation = 0.7

significance level = 0.05

Here at 95% CI, z value is +/- 1.96

Interval = mean +/- z*(s/sqrt(n))

substitute values

= 16 +/- 1.96*0.7/sqrt(33)

= 16 +/- 0.2388

Interval = (15.7612 , 16.2388)

P(Type II error) = P((15.7612 - 16.5)/(0.7/sqrt(33)) < (x-u)/(s/sqrt(n)) < (16.2388 - 16.5)/(0.7/sqrt(33)))

= P(-6.06 < z < -2.14)

= P(z < - 2.14) - P(z < - 6.06)

= 0.0161773 - 0 [since from z table]

= 0.0162

Now to give power overfilling by 0.5

Power = 1 - P(type II error)

= 1 - 0.0162

Power = 0.9838