Question

A 1977 survey of 100 randomly selected teenage girls between the ages of 13 and 15 showed that 32 of them had smoked in the past 12 months. A similar survey of 100 in 1993 showed that 62 of the girls had smoked in the past 12 months. (Use subscript 1 for 1977 and 2 for 1993). We want to test, at 5% level of significance, the hypothesis that proportion of smokers in 1977 is smaller than that of the 1993. Use the five-step hypothesis testing procedure.

Answer 1

To Test :-

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

p̂1 = 32 / 100 = 0.32
p̂2 = 62 / 100 = 0.62

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 = ( 32 + 62 ) / ( 100 + 100 )
p̂ = 0.47
q̂ = 1 - p̂ = 0.53
Z = ( 0.32 - 0.62) / √( 0.47 * 0.53 * (1/100 + 1/100) )
Z = -4.2503 ≈ - 4.25

Test Criteria :-
Reject null hypothesis if Z < -Z(α)
Z(α) = Z(0.05) = 1.645
Z < -Z(α) = -4.2503 < -1.645, hence we reject the null hypothesis
Conclusion :- We Reject H0

There is sufficient evidence to support the claim that proportion of smokers in 1977 is smaller than that of the 1993..