Question

For the population of middle-aged men who later develop diabetes mellitus, the distribution of baseline body mass indices is approximately normal with an unknown mean and standard deviation. A sample of 50men selected from this group has mean 25kg/m2and standard deviation 2.5. At the 0.05 level of significant, test whether the mean baseline body mass index for the population of middle-aged men who do develop diabetes is equal to 23kg/m2, the mean for the population of men who do not. Perform an appropriate two-sided hypothesis test

a)State the null and alternative hypotheses.

b)Which test statistic do we need to use? t or z? Why?

c)Calculate an appropriate test statistic. d)Calculate a p-value.

e)Can you reject H0at the 0.05 level of significance?

f)Interpret the result.

Answer 1

Part a)

To Test :-

H0 :- µ = 23
H1 :- µ ≠ 23

Part b)

We use t test, because we have sample standard deviation

Part c)

Test Statistic :-
t = ( X̅ - µ ) / ( S / √(n))
t = ( 25 - 23 ) / ( 2.5 / √(50) )
t = 5.6569
Test Criteria :-
Reject null hypothesis if | t | > t(α/2, n-1)
Critical value t(α/2, n-1) = t(0.05 /2, 50-1) = 2.01
| t | > t(α/2, n-1) = 5.6569 > 2.01
Result :- Reject null hypothesis

Decision based on P value
P - value = P ( t > 5.6569 ) = 0 i,e < 0.01
Reject null hypothesis if P value < α = 0.05 level of significance
P - value = 0 < 0.05 ,hence we reject null hypothesis
Conclusion :- Reject null hypothesis

Part e)

Conclusion :- Reject null hypothesis

Part f)

There is insufficient evidence to support the claim that the mean baseline body mass index for the population of middle-aged men who do develop diabetes is equal to 23kg/m2.