Question

A police officer expects that at least 60% of the vehicle stopped for speeding on Friday nights are under the influence of narcotics. A sample of 75 drivers who were stopped for speeding on a Friday night was taken. Fifty-two percent of the drivers in the sample were under the influence of narcotics. Perform the appropriate test at the 10% level of significance. Have you committed a type I error? Explain.

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:

Null hypothesis: H0: At least 60% of the vehicles stopped for speeding are under the influence of narcotics.

Alternative hypothesis: Ha: Less than 60% of the vehicles stopped for speeding are under the influence of narcotics.

H0: p ≥ 0.60 versus Ha: p < 0.60

This is a lower tailed test.

We are given

Level of significance = α = 0.10

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 = 75

p̂ = x/n = 0.52

p = 0.6

q = 1 - p = 0.4

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.52 - 0.6)/sqrt(0.6*0.4/75)

Z = -1.4142

Test statistic = -1.4142

P-value = 0.0786

(by using z-table)

P-value < α = 0.10

So, we reject the null hypothesis

There is not sufficient evidence to conclude that At least 60% of the vehicles stopped for speeding are under the influence of narcotics.

We committed a type I error because we reject the null hypothesis in the above test.