Question

A recent survey of households that put up Christmas trees polled both rural and urban households. Of the 148 rural households, 72 were found to put up a natural tree (as opposed to an artificial one) and of 270 urban households, 109 put up a natural tree. An association of Christmas tree growers wants to test the claim that a higher proportion of rural households prefer natural trees than urban households.

(a) State the claim, the negation of the claim, H0, and H1 (using equations and the parameters p 1and p 2).

(b) Find the p-value. Use α = .01to test the claim and state your conclusion about H0.

(c) State your conclusion about the original claim.

Answer 1

Part a)

H0 :- P1 = P2
H1 :- P1 > P2

Part b)

p̂1 = 72 / 148 = 0.4865
p̂2 = 109 / 270 = 0.4037
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 = ( 72 + 109 ) / ( 148 + 270 )
p̂ = 0.433
q̂ = 1 - p̂ = 0.567
Z = ( 0.4865 - 0.4037) / √( 0.433 * 0.567 * (1/148 + 1/270) )
Z = 1.6335

Decision based on P value
P value = P ( Z > 1.6335 ) = 0.0512
Reject null hypothesis if P value < α = 0.01
Since P value = 0.0512 > 0.01, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

Part c)

There is insufficient evidence to support the claim that a higher proportion of rural households prefer natural trees than urban households.