Question

The following measurements were recorded for the drying time, in hours, of a certain brand of latex paint: 3.4 2.5 4.8 2.9 3.6 2.8 3.3 5.6 3.7 2.8 4.4 4.0 5.2 3.0 4.8 Assuming that the measurements represent a random sample from a normal population, find the 95% confidence interval for the population mean.

Answer 1

Answer:

$3.79-2.14\frac{0.97}{\sqrt{15}}=3.25$    

$3.79+2.14\frac{0.97}{\sqrt{15}}=4.33$    

So on this case the 95% confidence interval would be given by (3.25;4.33)  

Step-by-step explanation:

Notation

$\bar X$ represent the sample mean for the sample  

$\mu$ population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size  

Solution to the problem

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 mean and the sample deviation we can use the following formulas:  

$\bar X= \sum_{i=1}^n \frac{x_i}{n}$ (2)  

$s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}$ (3)  

The mean calculated for this case is $\bar X=3.79$

The sample deviation calculated $s=0.97$

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

$df=n-1=15-1=14$

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(0.025,14)".And we see that $t_{\alpha/2}=2.14$

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

$3.79-2.14\frac{0.97}{\sqrt{15}}=3.25$    

$3.79+2.14\frac{0.97}{\sqrt{15}}=4.33$    

So on this case the 95% confidence interval would be given by (3.25;4.33)