Respuesta :
Answer:
The null hypothesis is [tex]H_0: p = 0.2[/tex]
The alternate hypothesis is [tex]H_a: p < 0.2[/tex]
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.
Step-by-step 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:
[tex]H_0: p = 0.2[/tex]
At the alternate hypothesis, we test if the proportion is less than 0.2, indicating a decline. So
[tex]H_a: p < 0.2[/tex]
The test statistic is:
[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
In which X is the sample mean, [tex]\mu[/tex] is the value tested at the null hypothesis, [tex]\sigma[/tex] is the standard deviation and n is the size of the sample.
20% is tested at the null hypothesis:
This means that [tex]\mu = 0.2, \sigma = \sqrt{0.2*0.8}[/tex]
A random sample of 100 adults over 40 years old found 15 who exercised at least twice a week.
This means that [tex]n = 100, X = \frac{15}{100} = 0.15[/tex]
Test statistic:
[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
[tex]z = \frac{0.15 - 0.2}{\frac{\sqrt{0.2*0.8}}{\sqrt{100}}}[/tex]
[tex]z = -1.25[/tex]
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.
