Question

. The expected mean of a normal population is 100, and its standard deviation is 12. A sample of 49 measurements gives a sample mean of 96. Using the α = 0.01 level of significance a test is to be made to decide between “population mean is 100” or “population mean is different than 100.” a) State null H0. b) What conclusion can be drawn at the given level of significance α = 0.01. c) What conclusion can be drawn if α = 0.05? d) What is the p-value of the test? e) State the type I and II errors. f) What is probability of type II error when, if mean μ really is 102 and α = 0.05 ?

Answer 1

solution:

the given information as follows:

population mean =

standard deviatin =

sample size = n = 49

sample mean =

significanc level =

null and alternative hypothesis

it is a two tailed test

test statistics:

p value = 2(value of z to the left of -2.33) = 2*0.0099 = 0.0198

b)

since p value 0.0198 > 0.01, so do not reject the null hypothesis

conclusion:

not rejecting the H0: at 0.01 significance level , concluded that there is not evidence that the mean is different from 100

c)

if

p value 0.0198 < 0.05, so rejecting the null hypothesis

conclusion:

rejecting the H0: at 0.05significanc level, concluded that there is enough evidence to support the claim that the mean is different from 100

d)

p value = 0.0198

e) if

critical value of z =

p(type II | = 102 ) =

=

= (value of z to the left of 0.79 ) - (value of z to the left of -3.13)

P(type II ) = 0.7852 - 0.0009 = 0.7843