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.

a)Conduct a test in P-value approach with the significance level of 0.05.

b) Suppose the researcher selected a random sample of size 50. 14 of 50 persons exposed to radiation felt discomfort. Conduct a test with the α=0.06 level of significance.

Answer 1

a)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.19
Alternative Hypothesis, Ha: p > 0.19

Rejection Region
This is right tailed test, for α = 0.05
Critical value of z is 1.64.
Hence reject H0 if z > 1.64

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.2182 - 0.19)/sqrt(0.19*(1-0.19)/220)
z = 1.07

P-value Approach
P-value = 0.1423
As P-value >= 0.05, fail to reject null hypothesis.

b)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.19
Alternative Hypothesis, Ha: p > 0.19

Rejection Region
This is right tailed test, for α = 0.06
Critical value of z is 1.55.
Hence reject H0 if z > 1.55

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.28 - 0.19)/sqrt(0.19*(1-0.19)/50)
z = 1.62

P-value Approach
P-value = 0.0526
As P-value < 0.06, reject the null hypothesis.