Questions: And randomly sampled group of patients at a major U.S. regional hospital became part of a nutrition study on dietary habits. Part of the study consisted of a 50-question survey asking about types of foods consumed. Each question was scored on a scale from one: most unhealthy behavior, to five: most healthy behavior. The answers were summed and averaged. The population of interest is the patients at the regional hospital. A prior survey of patients had found the mean score for the population of patients to be μ=2.9. After careful review of these data, the hospital nutritionist decided that patients could benefit from nutrition education. The current survey was implemented after patients were subjected to this education, and it produced these sample statistics for the 15 patients sampled: x̄=3.3 and s=1.2. We would like to know if the education improved nutrition behavior. We test the hypotheses H0: μ=2.9 versus Hα: μ>2.9.

And randomly sampled group of patients at a major U.S. regional hospital became part of a nutrition study on dietary habits. Part of the study consisted of a 50-question survey asking about types of foods consumed. Each question was scored on a scale from one: most unhealthy behavior, to five: most healthy behavior. The answers were summed and averaged. The population of interest is the patients at the regional hospital. A prior survey of patients had found the mean score for the population of patients to be μ=2.9. After careful review of these data, the hospital nutritionist decided that patients could benefit from nutrition education. The current survey was implemented after patients were subjected to this education, and it produced these sample statistics for the 15 patients sampled: x̄=3.3 and s=1.2. We would like to know if the education improved nutrition behavior. We test the hypotheses H0: μ=2.9 versus Hα: μ>2.9.

Transcript text: And randomly sampled group of patients at a major U.S. regional hospital became part of a nutrition study on dietary habits. Part of the study consisted of a 50-question survey asking about types of foods consumed. Each question was scored on a scale from one: most unhealthy behavior, to five: most healthy behavior. The answers were summed and averaged. The population of interest is the patients at the regional hospital. A prior survey of patients had found the mean score for the population of patients to be $\mu=2.9$. After careful review of these data, the hospital nutritionist decided that patients could benefit from nutrition education. The current survey was implemented after patients were subjected to this education, and it produced these sample statistics for the 15 patients sampled: $\bar{x}=3.3$ and $s=1.2$. We would like to know if the education improved nutrition behavior. We test the hypotheses $H_{0}: \mu=2.9$ versus $H_{\alpha}: \mu>2.9$.

Solution

Solution Steps

Step 1: Define the Hypotheses

We are testing the following hypotheses:

Null Hypothesis: $ H_0: \mu = 2.9 $
Alternative Hypothesis: $ H_{\alpha}: \mu > 2.9 $

Step 2: Calculate the t-Test Statistic

The t-test statistic is calculated using the formula:

\[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \]

Substituting the values:

\[ t = \frac{3.3 - 2.9}{1.2 / \sqrt{15}} \approx 1.2909944487358054 \]

Rounding to four significant digits, we have:

\[ t \approx 1.291 \]

Step 3: Conclusion

The calculated t-test statistic is $ t \approx 1.291 $. This value will be compared against the critical value from the t-distribution for $ n - 1 = 14 $ degrees of freedom at a chosen significance level (e.g., $ \alpha = 0.05 $) to determine if we reject the null hypothesis.