Question

6. 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

Answer)

A)

Null hypothesis Ho : u = 100

Alternate hypothesis Ha : u not equal to 100

As the population s.d is known here we can use standard normal z table to conduct the test.

Test statistics z = (sample mean - claimed mean)/(s d/√n)

Z = (96-100)/(12/√49) = -2.33

From z table, P(z<-2.33) = 0.0099

As the test is two tailed.

P-value = 2*0.0099 = 0.0198

B)

Since p-value is greater than the given significance 0.01.

Here we fail to reject the null hypothesis Ho.

We do not have enough evidence to conclude that mean is different from 100.

C)

Since p-value is less than the given significance 0.05.

Reject Ho

We have enough evidence to conclude that mean is different from 100.

D)

0.0198.