Question

A random sample is selected from a normal population with a mean mu equals 30 space a n d space s tan d a r d space d e v i a t i o n space sigma equals 8. After a treatment is administered to the individuals in the sample, the sample mean is found to be M=33. (a) If the sample consists of n=16 scores is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two tailed test with alpha space equals space 0.05. (b) If a sample consists of n= 64 scores, is the sample mean sufficient to conclude that the teatment has a significant effect? Use a two-tailed test with alpha space equals space 0.05. (c) Comparing your answers for parts a and b explain how the size of the sample influences the outcome of the hypothesis test?

Answer 1

Part a)

H0 :- µ = 30
H1 :- µ ≠ 30

Test Statistic :-
Z = ( X̅ - µ ) / ( σ / √(n))
Z = ( 33 - 30 ) / ( 8 / √( 16 ))
Z = 1.50

Test Criteria :-
Reject null hypothesis if | Z | > Z( α/2 )
Critical value Z(α/2) = Z( 0.05/2 ) = 1.96
| Z | > Z( α/2 ) = 1.5 < 1.96
Result :- Fail to reject null hypothesis

There is insufficient evidence to support the claim that  the treatment has a significant effect.

Part b)

Test Statistic :-
Z = ( X̅ - µ ) / ( σ / √(n))
Z = ( 33 - 30 ) / ( 8 / √( 64 ))
Z = 3.00

Test Criteria :-
Reject null hypothesis if | Z | > Z( α/2 )
Critical value Z(α/2) = Z( 0.05/2 ) = 1.96
| Z | > Z( α/2 ) = 3 > 1.96
Result :- Reject null hypothesis

There is sufficient evidence to support the claim that  the treatment has a significant effect.

Part c)

As the sample size increases, Z statistic also increases which lead to reject the null hypothesis i.e as sample size increases we will be colse to the decision that the treatment has a significant effect.