Question

Assume that you want to test the hypothesis that the mean weight of the male student population is 160 lbs.

If we assume that the standard deviation has a known value of 12 lbs and we want to test with a confidence level of 95%

a. What sample size should we use if we want one to have a probability of making a type II error of no more than 5% in the test when the actual mean is 162 lbs?

b. If the actual mean is 160 lbs, what is the probability that you conclude that the hypothesis is false?

c. If we want to use a 95% confidence interval to estimate the mean and determine that the estimation error should not be greater than 2 lbs, what sample size should we use?

d. If we decide to take 69 observations, what will be the probability of making a type II error if the actual mean is 163?

Answer 1

a)

Hypothesized mean μo=		160
true mean μa=		162
std deviation σ=		12.00
0.05 level critical Z=		1.96
0.05 level critical Zβ=		1.645
n=(Zα/2+Zβ)2σ2/(μo-μa)2=		468

b)

P( hypothesis is false given µ =160) =type I error =0.05

c)

for95% CI crtiical Z          =	1.96	from excel:normsinv(0.975)
standard deviation σ=	12.000
margin of error E =	2
required n=(zσ/E)2 =	139	Rounding up

d)

sample size    n=		69
std deviation σ=		12.00
std error ='σx=σ/√n=12/√69=		1.4446

rejection region: μ-Zα*σx <Xbar >μ+Zα*σx or 157.169<Xbar>162.831

P(Type II error) =P(157.169<Xbar<162.831|μ=163)=P(157.169-163)/1.445<z<(162.831-163)/1.445)=P(-4.04<z<-0.12)=0.4522