Question

A study is made of residents in Phoenix and its suburbs concerning the proportion of residents who subscribe to Sporting News. A random sample of n₁ = 86 urban residents showed that r₁ = 14 subscribed, and a random sample of n₂ = 97 suburban residents showed that r₂ = 20 subscribed. Does this indicate that a higher proportion of suburban residents subscribe to Sporting News? Use a 5% level of significance.

What is the value of the sample test statistic? (Test the difference p₁ − p₂. Do not use rounded values. Round your final answer to two decimal places.)

Find (or estimate) the P-value.

P-value > 0.2500.

125 < P-value < 0.250   

0.050 < P-value < 0.1250

.025 < P-value < 0.050

0.005 < P-value < 0.025

P-value < 0.005

Answer 1

Given that,

For urban residents : n1 = 86, r1 = 14 and p1^ = 14/86 = 0.1628

For suburban residents : n2 = 97, r2 = 20 and p2^ = 20/97 = 0.2062

Pooled proportion is,

p^ = (14 + 20) / (86 + 97) = 34/183 = 0.1858

The null and alternative hypotheses are,

H0 : p1 - p2 = 0

Ha : p1 - p2 < 0

This hypothesis test is a left-tailed test.

Test statistic is,

Z = (0.1628 - 0.2062) / √(0.1858 * (1 - 0.1858) * ((1/86) + (1/97)))

=> Z = -0.75

=> Test statistic = Z = -0.75

Using standard normal z-table we find the p-value as follows :

p-value = P(Z < -0.75) = 1 - P(Z < 0.75) = 1 - 0.7734 = 0.2266

=> p-value = 0.2266

Answer : 0.125 < P-value < 0.250