Question

1. a fitness expert claims that the proportion of all adults who meet their weekly exercise goals is 0.24. suspecting that the true population proportion is less than 0.24, a researcher surveys a random sample of 400 adults and finds that 90 of these adults meet their weekly exercise goals. based on this information, which one of the following statements is correct?

a. the test statistic should equal 0 because the sample proportion is almost exactly the same as the population proportion.
b. without knowing the chosen significance level, its impossible to obtain a p - value.
c. when a hypothesis test is conducted, the resulting test statistic should be smaller than 0 because 0.24 is larger than the sample proportion.
d. the test statistic will be larger than 0 because the claimed population proportion of 0.24 lies above the sample proportion.
e. the small difference between p and \\( \hat { p } \\) will yield a p - value close to 0 and strong evidence against the null hypothesis.

Explanation:

Step1: Calculate the sample proportion

The sample proportion \(\hat{p}=\frac{90}{400}=0.225\)

Step2: Recall the formula for the z - test statistic in a proportion test

The formula for the z - test statistic is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\), where \(p = 0.24\) (the hypothesized proportion), \(\hat{p}=0.225\) (the sample proportion), and \(n = 400\) (the sample size). Since \(\hat{p}

Step3: Analyze each option

• Option A: The sample proportion \(\hat{p}=0.225\) and \(p = 0.24\) are not the same. The test statistic \(z

eq0\).

• Option B: The \(P -\)value is calculated from the test statistic and the type of test (one - tailed or two - tailed), not from the significance level.
• Option C: Since \(\hat{p}
• Option D: Since \(\hat{p} 0\)
• Option E: The \(P -\)value is the probability of getting a test statistic as extreme or more extreme than the observed one. Since the difference \(\hat{p}-p=-0.015\) is not extremely large, the \(P -\)value is not close to \(0\)

Answer:

C. When a hypothesis test is conducted, the resulting test statistic should be smaller than 0 because 0.24 is larger than the sample proportion.