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 31 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)?
     

Answer 1

a)

null hypothesis:Ho μ		=	16
Alternate Hypothesis:Ha μ		≠	16
for 0.05 level with two tail test , critical z=			1.960	(from excel:normsinv(0.025)
Decision rule:reject Ho if absolute test stat|z|>1.96

sample size    n=		31
std deviation σ=		0.70
std error ='σx=σ/√n=0.7/√31=		0.1257

rejection region: μ-Zα*σx <Xbar >μ+Zα*σx or 15.754<Xbar>16.246

a) probability of making a Type II error :

P(Type II error) =P(15.754<Xbar<16.246|μ=16.5)=P(15.754-16.5)/0.126<z<(16.246-16.5)/0.126)=P(-5.93<z<-2.02)=0.0217