Question

A 99% confidence interval (in inches) for the mean height of a population is 65.7 < μ < 67.3. This result is based on a sample of size 144.
Construct the 95% confidence interval. (Hint: you will first need to find the sample mean and sample standard deviation).

Answer 1

Answer:

$66.5 -1.976* 0.3065= 65.89$

$66.5 +1.976* 0.3065= 67.11$

And the 95% confidence interval would be givne by (65.89; 67.11)

Step-by-step explanation:

For this case we have a 99% confidence interval for the true mean hegiht givne by:

$65.7 \leq \mu \leq 67.3$

And we can begin finding the mean with this formula:

$\bar X= \frac{65.7 +67.3}{2}= 66.5$

Now we can estimate the margin of error:

$ME = \frac{67.3-65.7}{2}= 0.8$

The sample size for this case is $n = 144$ then the degrees of freedom are given by:

$df = n-1= 144-1=143$

And then we can find a critical value for a 99% of confidence and 143 degrees of freedom using a significance level of 0.01 and we got:

$t_{\alpha/2}= 2.61$

Then the standard error is given by:

$SE = \frac{ME}{t_{\alpha/2}}= \frac{0.8}{2.61}= 0.3065$

Now we can find the other critical value for 95% of confidence and we got:

$t_{\alpha/2}= 1.976$

And the new confidence interval would be given by:

$66.5 -1.976* 0.3065= 65.89$

$66.5 +1.976* 0.3065= 67.11$

And the 95% confidence interval would be givne by (65.89; 67.11)