Question

In a random sample of 315 people 80 of them were smokers. Based on the above sample test whether the data provides sufficient evidence that the proportion of smokers is different from 20%, using 2% significance level. What type of error could you have made in your decision?

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 = 0.20 versus Ha: p ≠ 0.20

This is a two 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 = 80

n = sample size = 315

p̂ = x/n = 80/315 = 0.253968254

p = 0.2

q = 1 - p = 0.8

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.253968254 – 0.2)/sqrt(0.2*0.8/315)

Z = 2.3946

Test statistic = 2.3946

P-value = 0.0166

(by using z-table)

P-value < α = 0.02

So, we reject the null hypothesis

There is sufficient evidence to conclude that the proportion of smokers is different from 20%.