Question

Consider the following hypothesis test.

H₀: μ ≥ 10

H_a: μ < 10

The sample size is 125 and the population standard deviation is assumed known with σ = 5. Use α = 0.05.

(a) If the population mean is 9, what is the probability that the sample mean leads to the conclusion do not reject H₀?  (Round your answer to four decimal places.)

(b) What type of error would be made if the actual population mean is 9 and we conclude that H₀: μ ≥ 10 is true?

(c) What is the probability of making a type II error if the actual population mean is 8? (Round your answer to four decimal places. If it is not possible to commit a type II error enter NOT POSSIBLE.)

Answer 1

a)

for 0.05 level with left tail test , critical z=

-1.645

sample size    n=	125.00
std deviation σ=	5.000
std error ='σx=σ/√n=	0.4472

rejection region: Xbar <=μ+Zα*σx or Xbar<=10-1.645*0.4472 =

9.2643

P(Type II error) =P(Xbar>9.264|μ=9)=P(Z>(9.2643-9)/0.447)=P(Z>0.59)=

0.2776

(please try 0.2773 if this comes wrong)

b)

type II error , since we fail to reject a false null hypothesis

c)

P(Type II error) =P(Xbar>9.264|μ=8)=P(Z>(9.2643-8)/0.447)=P(Z>2.83)=

0.0023