Question

The head circumference (in cm) was recorded for each infant in a random sample of 25 full term newborn female infants. From the sample data, the mean head circumference was 34.12 cm. Assume that the distribution of head circumferences for full term newborn infants is approximately normal with a standard deviation of 1.18 cm. We wish to construct a 95% confidence interval for μ, the true mean head circumference of all full term newborn infants.What is the point estimate of μ? Enter your answer using two decimal places.

Answer 1

The point estimate of the mean is the sample mean. This value is already given: 34.12 cm.

In order to construct a 95% confidence interval for the mean, first let's calculate the value of z that corresponds to the value of alpha below:

$\begin{gathered} \frac{\alpha}{2}=100-95\\ \\ \frac{\alpha}{2}=5\\ \\ \alpha=2.5 \end{gathered}$

Looking at the z-table, the z-scores for a percentage of 2.5% and 97.5% are -1.96 and 1.96.

Now, let's use these values in the formula below to calculate the boundaries of the confidence interval:

$\begin{gathered} z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}\\ \\ \\ \\ -1.96=\frac{x_{lower}-34.12}{\frac{1.18}{5}}\\ \\ x_{lower}=34.12-1.96\cdot\frac{1.18}{5}\\ \\ x_{lower}=33.66\\ \\ \\ \\ 1.96=\frac{x_{upper}-34.12}{\frac{1.18}{5}}\\ \\ x_{upper}=34.12+1.96\cdot\frac{1.18}{5}\\ \\ x_{upper}=34.58 \end{gathered}$

Therefore the confidence interval is [33.66, 34.58].