Question

A physicist claims that more than 19% of all persons exposed to a certain amount of radiation will feel discomfort. A researcher selected a random sample, 48 of 220 persons exposed to radiation felt discomfort.

Identify the significance level. Check the assumptions to identify the sampling distribution under null hypothesis. Compute the P-value.

Answer 1

Solution:

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: About 19% of all persons exposed to a certain amount of radiation will feel discomfort.

Alternative hypothesis: Ha: More than 19% of all persons exposed to a certain amount of radiation will feel discomfort.

H0: p = 0.19 versus Ha: p > 0.19

This is an upper tailed test.

We assume

Level of significance = α = 0.05

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 = 48

n = sample size = 220

p̂ = x/n = 48/220 = 0.218181818

p = 0.19

q = 1 - p = 0.81

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.218181818 – 0.19)/sqrt(0.19*0.81/220)

Z = 1.0655

Test statistic = 1.0655

P-value = 0.1433

(by using z-table)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is not sufficient evidence to conclude that More than 19% of all persons exposed to a certain amount of radiation will feel discomfort.