Question

Consider the following random sample from a normal population: 14, 10, 13, 16, 12, 18, 15, and 11. What is the 95% confidence interval for the population variance?

Answer 1

Answer:

$13.625-2.365\frac{2.669}{\sqrt{8}}=11.393$  

$13.625+2.365\frac{2.669}{\sqrt{8}}=15.857$  

So on this case the 95% confidence interval would be given by (11.393;15.857)  

Step-by-step explanation:

Previous concepts  

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

Data: 14, 10, 13, 16, 12, 18, 15, 11

We can calculate the sample mean and deviation with the following formulas:

$\bar X = \frac{\sum_{i=1}^n X_i}{n}$

$s = \sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}$

$\bar X=13.625$ represent the sample mean  

$\mu$ population mean (variable of interest)  

s=2.669 represent the sample standard deviation  

n=8 represent the sample size  

Calculate the confidence interval

The confidence interval for the mean is given by the following formula:  

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$ (1)  

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:  

$df=n-1=8-1=7$  

Since the confidence is 0.95 or 95%, the value of $\alpha=0.05$ and $\alpha/2 =0.025$, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,7)".And we see that $t_{\alpha/2}=2.365$

Now we have everything in order to replace into formula (1):  

$13.625-2.365\frac{2.669}{\sqrt{8}}=11.393$  

$13.625+2.365\frac{2.669}{\sqrt{8}}=15.857$  

So on this case the 95% confidence interval would be given by (11.393;15.857)