Question

The proportion of online shoppers who actually make a purchase appears to be relatively constant over time. In 2013, among a sample of 388 online shoppers, 160 purchased merchandise. In 2017, for a sample of 307 online shoppers, 144 purchased merchandise. At the 0.05 level of significance, did the proportion of online shoppers change from 2013 to 2017?

State the null and alternate hypotheses.

H0: π2013	(Click to select)  =  >  <  ≠	π2017
H1: π2013	(Click to select)  =  >  <  ≠	π2017

Make the decision rule. (Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.)

Evaluate the test statistic. (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.)

What is the decision regarding the null hypothesis?

Answer 1

To Test :-

H0 :- P1 = P2
H1 :- P1 ≠ P2

Test Criteria :-

Reject null hypothesis if Z < -Z(α/2)
Z(α/2) = Z(0.05/2) = 1.96
Z < - 1.96 OR Z > 1.96

p̂1 = 160 / 388 = 0.4124
p̂2 = 144 / 307 = 0.4691

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 = ( 160 + 144 ) / ( 388 + 307 )
p̂ = 0.4374
q̂ = 1 - p̂ = 0.5626
Z = ( 0.4124 - 0.4691) / √( 0.4374 * 0.5626 * (1/388 + 1/307) )
Z = -1.50

Z > -Z(α/2) = -1.4959 > -1.96, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0