Question

Approximately 10% of the US population smoke cigarettes. A local county believed their community has a lower smoke rate and commissioned a survey of 400 randomly selected individuals. The survey found that 10 of the 400 participants smoke cigarettes. Allow a 2% significance level. What decision will you arrive at regarding the 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:

Null hypothesis: H0: A community has a smoke rate of 10%.

Alternative hypothesis: Ha: A community has a smoke rate less than 10%.

H0: p = 0.10 versus Ha: p < 0.10

This is a lower tailed test.

We are given

Level of significance = α = 0.02

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

x = number of items of interest = 10

n = sample size = 400

p̂ = x/n = 10/400 = 0.025

p = 0.1

q = 1 - p = 0.9

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.025 - 0.1)/sqrt(0.1*0.9/400)

Z = -5.0000

Test statistic = -5.0000

P-value = 0.0000

(by using z-table)

P-value < α = 0.02

So, we reject the null hypothesis

There is sufficient evidence to conclude that a community has a smoke rate less than 10%.