Question

A May 8, 2008, report on National Public Radio (www.npr.org) noted that the average age of first- time mothers in the United States is slightly higher than 25 years. Suppose that a recently taken random sample of 57 first-time mothers from Missouri produced an average age of 23.90 years and that the population standard deviation is known to be 4.80 years.

a. Find the p-value for the test of hypothesis with the alternative hypothesis that the current mean age of all first-time mothers in Missouri is less than 25 years. Will you reject the null hypothesis at α = .05?

b. Test the hypothesis of part a using the critical-value approach and α = .05.

Answer 1

Part a)

To Test :-
H0 :- µ = 25
H1 :- µ < 25

Test Statistic :-
Z = ( X - µ ) / ( σ / √(n))
Z = ( 23.9 - 25 ) / ( 4.8 / √( 57 ))
Z = -1.7302

Decision based on P value
Reject null hypothesis if P value < α = 0.05 level of significance
P value = P ( Z < 1.7302 ) = 0.0418 ( From Z table )
Since 0.0418 < 0.05 ,hence we reject null hypothesis
Result :- Reject null hypothesis

Part b)

Test Criteria :-
Reject null hypothesis if Z < -Z(α)
Critical value Z(α) = Z(0.05) = 1.645 ( from Z table )
Z < -Z(α) = -1.7302 < -1.645
Result :- Reject null hypothesis

There is sufficient evidence to support the claim  that the current mean age of all first-time mothers in Missouri is less than 25 years.