Question

7. The survey took a poll that asked 352employed people aged 25 to 40 whether they used a computer at work, and 264said they did. They alsoasked the same question of 504employed people aged 41 to 65, and 390of them said that they used a computer at work. Can you conclude that the proportion who use a computer at work is less among those aged 25 to 40 than among those aged 41 to 65? Use the α= 0.05.

Answer 1

To Test :-

H0 :- P1 = P2
H1 :- P1 ≠ P2

p̂1 = 264 / 352 = 0.75
p̂2 = 390 / 504 = 0.7738

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 = ( 264 + 390 ) / ( 352 + 504 )
p̂ = 0.764
q̂ = 1 - p̂ = 0.236
Z = ( 0.75 - 0.7738) / √( 0.764 * 0.236 * (1/352 + 1/504) )
Z = -0.8073

Test Criteria :-
Reject null hypothesis if Z < -Z(α)
Z(α) = Z(0.05) = 1.645
Z > -Z(α) = -0.8073 > -1.645, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

Decision based on P value
P value = P ( Z < -0.8073 ) = 0.7902
Reject null hypothesis if P value < α = 0.05
Since P value = 0.7902 > 0.05, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

There is insufficient evidence to support the claim that the proportion who use a computer at work is less among those aged 25 to 40 than among those aged 41 to 65.