Question

question 7 1 pts 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? the test statistic should equal 0 because the sample proportion is almost exactly the same as the population proportion. without knowing the chosen significance level, its impossible to obtain a p - value. when a hypothesis test is conducted, the resulting test statistic should be smaller than 0 because 0.24 is larger than the sample proportion. the test statistic will be larger than 0 because the claimed population proportion of 0.24 lies above the sample proportion. 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 sample proportion

Sample proportion $\hat{p}=\frac{90}{400}=0.225$

Step2: Analyze test statistic formula

The formula for the test statistic in a one - sample proportion test is $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$. Here, $p = 0.24$, $\hat{p}=0.225$, $n = 400$. Since $\hat{p}0$. So, $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}<0$

Step3: Analyze each option

• First option:

The sample proportion $\hat{p}=0.225
eq p = 0.24$, so the test statistic is not $0$.

• Second option:

The $P - value$ is calculated based on the test statistic (which is calculated from sample data and hypothesized proportion), not the significance level.

• Third option:

As shown in Step 2, since $\hat{p}

• Fourth option:

Since $\hat{p}

• Fifth option:

The difference $\hat{p}-p=-0.015$. The $P - value$ is the probability of getting a test statistic as extreme or more extreme than the observed one. A small difference (in absolute value) will yield a $P - value$ close to $0.5$ (not close to $0$) if the sample size is not extremely large

Answer:

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