Answer:
The 95% confidence interval for the true difference between the mean birth weights for "zinc" babies versus "placebo" babies is (-515.8566, -192.1434)
Step-by-step explanation:
Let [tex]\mu_{1}-\mu_{2}[/tex] be the true difference between the mean birth weights for "zinc" babies versus "placebo" babies. We have the sample sizes [tex]n_{1} = 172[/tex] babies and [tex]n_{2} = 158[/tex] babies, the unbiased point estimate for [tex]\mu_{1}-\mu_{2}[/tex] is [tex]\bar{x}_{1} - \bar{x}_{2}[/tex], i.e., 3072-3426 = -354
The standard error is given by [tex]\sqrt{\frac{\sigma^{2}_{1}}{n_{1}}+\frac{\sigma^{2}_{2}}{n_{2}}}[/tex], i.e.,
[tex]\sqrt{\frac{(792)^{2}}{172}+\frac{(708)^{2}}{158}}[/tex] = 82.5799. Then, the endpoints for a 95% confidence interval for [tex]\mu_{1}-\mu_{2}[/tex] is given by
-354-[tex](z_{0.05/2})[/tex]82.5799 and -354+[tex](z_{0.05/2})[/tex]82.5799, i.e.,
-354-[tex](z_{0.025})[/tex]82.5799 and -354+[tex](z_{0.025})[/tex]82.5799 where [tex]z_{0.025}[/tex] is the 2.5th quantile of the standard normal distribution, i.e., -1.96, so, we have
-354-(1.96)(82.5799) and -354+(1.96)(82.5799), i.e.,
-515.8566 and -192.1434
