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The t-test is a statistical method that helps us determine if there is a significant difference between the means of two groups. In this case, we're examining whether the average zinc intake for males aged 65-74 is different from the recommended 15 mg/day.
To perform a t-test, we first set our null hypothesis (\( H_0 \)) stating that the average intake is equal to the recommendation (\( \mu = 15 \)). Next, we have our alternative hypothesis (\( H_a \)), stating it is less than the recommendation (\( \mu < 15 \)).
We calculate the t-statistic using the formula:

• \( t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \)

where \( \bar{x} \) is the sample mean, \( \mu \) is the population mean under the null hypothesis, \( s \) is the standard deviation, and \( n \) is the sample size. This statistic tells us how far off our sample is from the null hypothesis in standard error units.

The critical value is a threshold that determines whether the test statistic is extreme enough to reject the null hypothesis. It depends on the significance level (\( \alpha \)), which is commonly set at 0.05 (5%). This significance level represents the probability of rejecting the null hypothesis when it is actually true.
For a one-tailed t-test with 114 degrees of freedom in this study, the critical value can be obtained from a t-distribution table or using statistical software. Here, the critical value is approximately \( -1.658 \).
When the calculated t-statistic is compared to the critical value, it helps determine the decision about our hypotheses.

• If the t-statistic is more extreme than the critical value, we reject the null hypothesis.
• If it is not, we fail to reject the null hypothesis.

This zinc intake study investigates whether older males' average zinc consumption is less than the recommended daily allowance of 15 mg. Such studies are crucial in identifying nutritional deficiencies among specific age groups.
The study gathered data from a sample of 115 males aged 65-74, reporting an average zinc intake (\( \bar{x} \)) of 11.3 mg with a standard deviation (\( s \)) of 6.43 mg. These parameters help perform the t-test for statistical analysis.
The outcomes from this study can inform dietary recommendations, policies, and individual health advice to potentially mitigate health problems related to inadequate zinc intake.

Statistical significance refers to the likelihood that an observed outcome is not due to chance. In hypothesis testing, we declare statistical significance if our test statistic falls in the rejection region defined by the critical value.
In this zinc intake study, we found the t-statistic to be \( -6.18 \), while the critical value was \( -1.658 \). Since \( -6.18 < -1.658 \), the result is statistically significant, suggesting that the average zinc intake among males aged 65-74 is significantly lower than the recommended 15 mg.
Achieving statistical significance implies confidence in our findings, but it is essential to consider the practical significance, or the real-world implications of these results.