According to the CDC 494 of American adults 18 years old and

According to the CDC, 49.4% of American adults 18 years old and over meet their Physical Activity Guidelines for aerobic activity. The owners of a gym want to know if the proportion of their members who met these requirements is different from the national average. They took a random sample of 40 members and 28 met the CDC’s guidelines. Perform a hypothesis test to answer their question. (Use a 5% significance level).

- Show all the steps, starting with your hypotheses all the way to your final conclusion.

- If the results are significant, calculate a 95% confidence interval for the true proportion

Solution

According to the CDC, 49.4% of American adults 18 years old and over meet their Physical Activity Guidelines for aerobic activity. The owners of a gym want to know if the proportion of their members who met these requirements is different from the national average. They took a random sample of 40 members and 28 met the CDC’s guidelines. Perform a hypothesis test to answer their question. (Use a 5% significance level).

- Show all the steps, starting with your hypotheses all the way to your final conclusion.

- If the results are significant, calculate a 95% confidence interval for the true proportion

H_0: P = 0.494 \\ \\ \\ H_1: P \ eq   0.494

P=28/40=0.7

z = \\frac{ p - P}{\\sqrt{ P*(1-P)/n}}

z = \\frac{ 0.7 - 0.44}{\\sqrt{ 0.454 *(1- 0.454)/40}}

= 2.6059

Table value of z at 0.05 level = ( -1.96, 1.96) ( two tailed test)

Calculated z = 2.6059 falls in the rejection region.

The null hypothesis is rejected.

We conclude that the proportion of their members who met these requirements is different from the national average. .

95% Confidence Interval Estimate for the Proportion

Data

Sample Size

40

Number of Successes

28

Confidence Level

95%

Intermediate Calculations

Sample Proportion

0.7

Z Value

-1.9600

Standard Error of the Proportion

0.0725

Interval Half Width

0.1420

Confidence Interval

Interval Lower Limit

0.5580

Interval Upper Limit

0.8420