Question

Childhood obesity: A national health survey weighed a sample of 542 boys aged 6-11 and found that 77 of them were overweight. They weighed a sample of 473 girls aged 6-11 and found that 80 of them were overweight. Can you conclude that the proportion of boys who are overweight is less than the proportion of girls who are overweight? Let p1 denote the proportion of boys who are overweight and p2 denote the proportion of girls who are overweight. Use the =α0.10 level of significance and the P-value method.

a.) State the null and alternate hypotheses.

(b) Compute the test statistic.

c.) Compute the P-value.

d.) Determine whether to reject H0.

e.) State the conclusion

.

Answer 1

Part a)

H0 :- P1 = P2
H1 :- P1 < P2

Part b)

p̂1 = 77 / 542 = 0.1421
p̂2 = 80 / 473 = 0.1691

Test Statistic :-
Z = ( p̂1 - p̂2 ) / √(p̂ * q̂ * (1/n1 + 1/n2) ) )
p̂ is the pooled estimate of the proportion P
p̂ = ( x1 + x2) / ( n1 + n2)
p̂ = ( 77 + 80 ) / ( 542 + 473 )
p̂ = 0.1547
q̂ = 1 - p̂ = 0.8453
Z = ( 0.1421 - 0.1691) / √( 0.1547 * 0.8453 * (1/542 + 1/473) )
Z = -1.1896

Part c)

P value = P ( Z < -1.1896 ) = 0.8829

Part d)

Reject null hypothesis if P value < α = 0.1
Since P value = 0.8829 > 0.1, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

Part e)

There is insufficient evidence to support the claim that the proportion of boys who are overweight is less than the proportion of girls who are overweight.