Paper For Above instruction
Introduction
The ongoing debate between the Food and Drug Administration (FDA) and the pharmaceutical company (ntbnP) revolves around whether the dietary supplement provided by ntbnP delivers the claimed amount of cobalamine (Vitamin B12) to teenagers. To assess this, statistical hypothesis testing is employed to evaluate whether the mean cobalamine level in teens aligns with the RDA of 2.4 µg or is significantly lower, indicating that teens are not receiving sufficient amounts. Given the data collected and the assumptions made about population standard deviation, a one-sample z-test for the mean is appropriate for this analysis.
Choice of Hypothesis Test
Since the population standard deviation (σ) is known (0.56 µg), and the sample size is relatively small (n=10), a z-test is suitable for this scenario. A z-test is used when σ is known; otherwise, a t-test would be necessary. The z-test allows us to determine whether the sample mean significantly differs from the hypothesized population mean of 2.4 µg, considering the known variability in the population.
Formulation of Hypotheses
The null hypothesis (H■) posits that the true mean cobalamine level in teens is equal to the RDA, i.e., H■: µ = 2.4 µg. The alternative hypothesis (H■) is that the mean cobalamine level is less than the RDA, i.e., H■: µ < 2.4 µg. This specifies a left-tailed test, as the concern is whether teens are under-consuming vitamin B12 relative to the RDA.
Calculation of Test Statistic
The sample mean (\( \bar{x} \)) is calculated as:
\( \bar{x} = \frac{1.85 + 2.35 + 1.87 + 1.90 + 1.37 + 2.35 + 2.55 + 2.28 + 1.95 + 2.49}{10} = \frac{20.97}{10} = 2.097 \) µg.
The test statistic (z) is computed using:
\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} = \frac{2.097 - 2.4}{0.56 / \sqrt{10}} = \frac{-0.303}{0.177} \approx -1.71 \]
Determining the p-value
Using standard normal distribution tables or software, the p-value corresponding to z = -1.71 in a left-tailed test is approximately 0.043. Since this p-value is less than the common significance level (α = 0.05), we reject the null hypothesis, indicating significant evidence that the true mean dose is less than 2.4 µg.
Interpretation and Conclusions
Given the p-value of approximately 0.043, which is less than 0.05, we reject H■. This suggests that teens, based on the sample data, are likely receiving less than the RDA of cobalamine, and ntbnP's claim that their supplement provides the full RDA may be unfounded. From an FDA perspective, evidence supports the claim that teenagers are not being adequately supplemented.
If I were representing ntbnP, I might argue that the sample size is small, and the result is borderline; thus, further testing with a larger sample is necessary. Conversely, if I were representing the FDA, I would emphasize the statistical evidence indicating that the supplement does not meet the RDA, reinforcing the need for regulatory action or further investigation.
What if Population Standard Deviation Is Unknown?
If the population standard deviation was unknown, the appropriate approach would involve conducting a t-test instead of a z-test. The t-test accounts for uncertainty in estimating variability from the sample data, especially with small sample sizes. The test statistic would then be:
\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]
where s is the sample standard deviation. Degrees of freedom would be \( n - 1 = 9 \), and the p-value would be obtained from the t-distribution table or software accordingly.
Conclusion
In summary, based on the available data and assumptions, a z-test indicates that the mean level of cobalamine in teens is significantly less than the RDA, supporting the claim that teens are under-supplemented by ntbnP's product. The decision hinges on the significance level and p-value, and the approach would change if the population variance was unknown, favoring a t-test.
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