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Regardless of age, about 20% of American adults participate in fitness activities at least twice a week. A random sample of 100 adults over 40 years old found 15 who exercised at least twice a week. Is this evidence of a decline in participation after age 40? Use α = .05. a) Write test hypothesis b) Compute test statistic. c) Apply any method to draw conclusion.

User Dga
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Answer:

The null hypothesis is
H_0: p = 0.2

The alternate hypothesis is
H_a: p < 0.2

The p-value of the test is 0.1056 > 0.05, which means that there is not evidence of decline in participation after age 40 using α = 0.05.

Explanation:

20% of American adults participate in fitness activities at least twice a week. Test if there is evidence of a decline in participation after age 40.

At the null hypothesis, we test that the proportion is of 0.2, so:


H_0: p = 0.2

At the alternate hypothesis, we test if the proportion is less than 0.2, indicating a decline. So


H_a: p < 0.2

The test statistic is:


z = (X - \mu)/((\sigma)/(√(n)))

In which X is the sample mean,
\mu is the value tested at the null hypothesis,
\sigma is the standard deviation and n is the size of the sample.

20% is tested at the null hypothesis:

This means that
\mu = 0.2, \sigma = √(0.2*0.8)

A random sample of 100 adults over 40 years old found 15 who exercised at least twice a week.

This means that
n = 100, X = (15)/(100) = 0.15

Test statistic:


z = (X - \mu)/((\sigma)/(√(n)))


z = (0.15 - 0.2)/((√(0.2*0.8))/(√(100)))


z = -1.25

P-value of the test:

The p-value of the test is the probability of finding a sample proportion below 0.15, which is the p-value of z = -1.25.

Looking at the z-table, z = -1.25 has a pvalue of 0.1056.

The p-value of the test is 0.1056 > 0.05, which means that there is not evidence of decline in participation after age 40 using α = 0.05.

User Ppiotrowicz
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