Question

The following hypotheses are given. H0: p ≤ 0.83 H1: p > 0.83 A sample of 128 observations revealed that = 0.73. At the 0.05 significance level, can the null hypothesis be rejected?

a. State the decision rule. (Round the final answer to 3 decimal places.) Reject CorrectH0 and and accept CorrectH1 if z>... or z<

Answer 1

Here, we have to use one sample z test for the population proportion.

The null and alternative hypotheses for this test are given as below:

H₀: p ≤ 0.83 versus H_a: p > 0.83

This is an upper tailed test.

We are given

Level of significance = α = 0.05

Critical Z value = 1.6449

(by using z-table)

Decision rule: Reject H₀ if test statistic Z > 1.645

Test statistic formula for this test is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size

n = sample size = 128

p̂ = x/n = 0.73

p = 0.83

q = 1 - p = 0.17

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.73 - 0.83)/sqrt(0.83*0.17/128)

Z = -3.0119

Test statistic = -3.0119

P-value = 0.9987

(by using z-table)

P-value > α = 0.05

So, we do not reject the null hypothesis.