Question

You are given the following hypotheses: Null hypothesis: p = 0.3
Alternative hypothesis: ? ≠ 0.30

You decide to take a sample of size 90. Suppose we will reject the null hypothesis if the probability of an outcome as surprising as ?̂ occurring is less than 5%. (i.e., a “p-value” of .05). What values ?̂ would cause us to reject the null hypothesis? Hint: Your answer should be “if ?̂ is anything bigger than ____ or anything smaller than____.”

Answer 1

Given that, the null and alternative hypotheses are,

H0 : p = 0.30

Ha : p ≠ 0.30

sample size (n) = 80

p-value = 0.05

We want to find, the sample proportions and such that,

First we find, the z-score such that, P(Z < -z) + P(Z > z) = 0.05

P(Z < -z) + P(Z > z) = 0.05

=> [ 1 - P(Z < z) ] + [ 1 - P(Z < z) ] = 0.05

=> 2 - 2 * P(Z < z) = 0.05

=> 2 * [ 1 - P(Z < z) ] = 0.05

=> 1 - P(Z < z) = 0.025

=> P(Z < z) = 0.975

Using standard normal z-table we get z-score corresponding probability of 0.975 is, z = 1.96

=> P(Z < -1.96) + P(Z > 1.96) = 0.05

Therefore,

For z = -1.96

For z = 1.96

Hence,

Answer : if is anything bigger than 0.3947 or anything smaller than 0.2053, we should reject the null hypothesis.