Question

A marketing survey involves product recognition in New York and California. Of 932 New Yorkers surveyed, 285 knew the product while 602 out of 147 Californians knew the product. At the 11% significance level, use the critical value method to test the claim that the recognition rates are the same in both states. Use table

Answer 1

p1cap = X1/N1 = 285/932 = 0.3058
p1cap = X2/N2 = 147/602 = 0.2442
pcap = (X1 + X2)/(N1 + N2) = (285+147)/(932+602) = 0.2816

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 ≠ p2

Rejection Region
This is two tailed test, for α = 0.11
Critical value of z are -1.6 and 1.6.
Hence reject H0 if z < -1.6 or z > 1.6

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.3058-0.2442)/sqrt(0.2816*(1-0.2816)*(1/932 + 1/602))
z = 2.62

Reject the H0
the recognition rates are not the same in both states