Question

About 20% of Americans participate in fitness activities at least twice a week. A researcher would like to know if the participation rate is lower for older adults. A simple random sample of 180 adults over the age of 50 shows that 27 participate in a fitness activity at least twice a week. Is there sufficient evidence to conclude that the proportion of adults over the age of 50 who participate in fitness activities at least twice a week is less than 20%? Use the α = 0.05 level of significance.
(a) Set up the null and alternative hypotheses.
(b) Verify that the conditions to conduct a test are satisfied.
(c) Compute the test statistic. Round your answer to two decimal places.
(d) Determine the P-value. Round your answer to four decimal places.
(e) Make a decision and state the conclusion.
(f) What would it mean, in the context of this problem, to make a Type I error?

Answer 1

a)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.2
Alternative Hypothesis, Ha: p < 0.2

b)
conditions are satisfied,
np > 5 and nq > 5

c)
Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.15 - 0.2)/sqrt(0.2*(1-0.2)/180)
z = -1.68

d)
P-value Approach
P-value = 0.0465

e)
As P-value < 0.05, reject the null hypothesis.

f)
type I error will result into the conclusion that the propotion is les than 0.2, however in actual it is not.