To compare the placebo and zinc groups, we can perform a two-sample t-test to determine if the difference in means is statistically significant. Here’s a breakdown of the calculation:

Step 1: Define Hypotheses

1. Null Hypothesis (H0): There is no difference in means between the placebo and zinc groups ($\mu_{placebo} = \mu_{zinc}$).
2. Alternative Hypothesis (H1): There is a difference in means between the placebo and zinc groups ($\mu_{placebo} \neq \mu_{zinc}$).

Step 2: Calculate the Test Statistic

For a two-sample t-test, the formula for the test statistic $t$ is: $t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{N_1} + \frac{s_2^2}{N_2}}}$ where:

• $\bar{X}_1 = 8.1$ (mean of placebo)
• $\bar{X}_2 = 4.5$ (mean of zinc)
• $s_1 = 1.8$ (standard deviation of placebo)
• $s_2 = 1.6$ (standard deviation of zinc)
• $N_1 = 23$ (sample size of placebo)
• $N_2 = 25$ (sample size of zinc)

Calculation:

1. Difference in means: $\bar{X}_1 - \bar{X}_2 = 8.1 - 4.5 = 3.6$

2. Pooled standard error: $\text{SE} = \sqrt{\frac{1.8^2}{23} + \frac{1.6^2}{25}} = \sqrt{\frac{3.24}{23} + \frac{2.56}{25}}$

3. Calculate each part:

   • $\frac{3.24}{23} \approx 0.1409$
   • $\frac{2.56}{25} \approx 0.1024$

4. Total standard error: $\text{SE} = \sqrt{0.1409 + 0.1024} = \sqrt{0.2433} \approx 0.4933$

5. t-value: $t = \frac{3.6}{0.4933} \approx 7.30$

Step 3: Determine Degrees of Freedom

Using the Satterthwaite approximation for degrees of freedom $df$: $df \approx \frac{\left(\frac{s_1^2}{N_1} + \frac{s_2^2}{N_2}\right)^2}{\frac{\left(\frac{s_1^2}{N_1}\right)^2}{N_1 - 1} + \frac{\left(\frac{s_2^2}{N_2}\right)^2}{N_2 - 1}}$

Substituting values will yield an approximate $df$. We then compare our $t$-value to the critical value or obtain a $p$-value to assess significance.

Would you like a detailed continuation with exact degrees of freedom, or interpretation based on a p-value approach?

---

1. What is the significance level (e.g., 0.05 or 0.01) you’d like for this test?
2. Are you interested in one-sided or two-sided test results?
3. Would you like further explanation on the Satterthwaite approximation?
4. How would we interpret a significant result in this context?
5. Would you like to see this calculated using Python or statistical software?

Tip: When comparing two groups, checking assumptions like normality and equal variances can strengthen the validity of a t-test.