Question

Historically emergency responders respond in less than 5 minutes 74% of the time. A recent sample of n = 715 found 70% of emergency calls had a response time of less than 5 minutes. Perform a hypothesis test with α = 2.5% to see if the response time percentage has significantly decreased.

A. What is H_o?

B. What is H₁?

C. What is the value of the test statistic?

D. What is the value of the critical value?

E. What is the conclusion to this hypothesis test?

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:

(A)

Null hypothesis: H0: The response time percentage has not significantly decreased.

H0: p = 0.74

Versus

(B)

Alternative hypothesis: Ha: The response time percentage has significantly decreased.

Ha: p < 0.74

This is a lower tailed test.

We are given

Level of significance = α = 0.025

(C)

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 = 715

p̂ = x/n = 0.70

p = 0.74

q = 1 - p = 0.26

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.70 – 0.74)/sqrt(0.74*0.26/715)

Z = -2.4384

Test statistic = -2.4384

(D)

Critical value = -1.96

P-value = 0.0074

(by using z-table)

(E)

Test statistic < Critical value

P-value < α = 0.025

So, we reject the null hypothesis

There is sufficient evidence to conclude that the response time percentage has significantly decreased.