Question

In order to help identify baby growth patterns that are unusual, we need to construct a confidence interval of the mean head circumference of all babies that are two months old. A random sample of 100 babies is obtained, and the mean head circumference is found to be 40.6 cm. Assuming that the population standard deviation is known to be 1.6 cm, find a 99% confidence interval estimate of the mean head circumference of all 2-month-old babies.

Answer 1

Answer:

The 99% confidence interval estimate of the mean head circumference of all 2-month-old babies is (40.18cm, 41.02cm).

Step-by-step explanation:

Our sample size is 100.

The first step to solve this problem is finding our degrees of freedom, that is, the sample size subtracted by 1. So

$df = 100-1 = 99$.

Then, we need to subtract one by the confidence level $\alpha$ and divide by 2. So:

$\frac{1-\alpha}{2} = \frac{0.01}{2} = 0.005$

Now, we need our answers from both steps above to find a value T in the t-distribution table. So, with 99 and 0.005 in the t-distribution table, we have $T = 2.626$

Now, we find the standard deviation of the sample. This is the division of the standard deviation by the square root of the sample size. So

$s = \frac{1.6}{\sqrt{100}} = 0.16$

Now, we multiply T and s

$M = Ts = 2.626*0.16 = 0.42$

For the lower end of the interval, we subtract the sample mean by M. So the lower end of the interval here is

$L = 40.6 - 0.42 = 40.18$cm

For the upper end of the interval, we add the sample mean and M. So the upper end of the interval here is

$L = 40.6 + 0.42 = 41.02$cm

So

The 99% confidence interval estimate of the mean head circumference of all 2-month-old babies is (40.18cm, 41.02cm).