Mean birthweight is studied because low birthweight is an indicator of infant mortality. A study of babies in Norway published in the International Journal of Epidemiology shows that birthweight of full-term babies (37 weeks or more of gestation) are very close to normally distributed with a mean of 3600 g and a standard deviation of 600 g. Suppose that Melanie is a researcher who wishes to estimate the mean birthweight of full-term babies in her hospital. What is the minimum number of babies should she sample if she wishes to be at least 95% confident that the mean birthweight of the sample is within 100 grams of the the mean birthweight of all babies? Assume that the distribution of birthweights at her hospital is normal with a standard deviation of 600 g.

For this case, we need the value of n such that Melanie is 95% confident that the mean birthweight of the sample is within 100 grams of the mean of the birthweight of all babies.

This 100 grams difference from mean is the margin of error as this is deciding the confidence's interval's endpoints from the mean.

The margin of error is [tex]MOE = Z_{\alpha /2}\dfrac{\sigma}{\sqrt{n}}[/tex] (we'll use population standard deviation here)

At 95% confidence, the level of significance is 100% - 95% = 5% = 0.05. The critical value [tex]Z_{\alpha/2} = \pm 1.96[/tex]

Since the standard deviation is of 600 grams, and margin of error needed is 100 g or low (as it is needed to be within 100 g) width, thus, we get the value of n as:

[tex]MOE \leq Z_{\alpha /2}\dfrac{\sigma}{\sqrt{n}}\\\\100 \leq \pm 1.96 \times \dfrac{600}{\sqrt{n}}\\\\n \leq (11.76)^2 = 138.29 \approx 139[/tex] (an integer)

Thus, the minimum number of babies should she sample for getting desired confidence interval is 139 approx

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