A production line operation is tested for filling weight accuracy using the foll

Question

A production line operation is tested for filling weight accuracy using the following hypotheses. Hypothesis Conclusion and Action H0: = 16 Filling okay, keep running Ha: 16 Filling off standard; stop and adjust machine The sample size is 34 and the population standard deviation is = 0.9. Use = .05.

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

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

Explanation / Answer

let Xbar be the sample mean. let the population mean be u

so Xbar follows a normal distribution with mean u and standard deviation=0.9/sqrt(34)

level of significance=alpha=0.05

H0: u=16 vs H1: u>16

the test statistic is given by T=(Xbar-16)*sqrt(34)/0.9 which under H0 follows N(0,1)

H0 is rejected iff t>tao0.05

where t is the observed value of T and tao0.05 is the upper 0.05 point of a N(0,1) distribution.

from R, tao0.05=1.64

so H0 is rejected when (Xbar-16)*sqrt(34)/0.9>1.64

or, Xbar>16+1.64*0.9/sqrt(34)=16.25

a) the machine is overfilling by .5 ounces

P[type 2 error]=P[accepting H0 | Ha is true]

=P[Xbar<16.25] where Xbar follows normal with mean 16.5 and standard deviation=0.9/sqrt(34)

=P[(Xbar-16.5)*sqrt(34)/0.9<(16.25-16.5)*sqrt(34)/0.9]=P[Z<-1.6197]=0.0526 [answer] [Z~N(0,1)]

b) so power of the test is 1-P[type 2 error]=1-0.0526=0.9474 [answer]

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