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

`3.79-2.14(0.97)/(√(15))=3.25`

`3.79+2.14(0.97)/(√(15))=4.33`

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

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)(s)/(√(n))` (1)

In order to calculate the mean and the sample deviation we can use the following formulas:

`\bar X= \sum_(i=1)^n (x_i)/(n)` (2)

`s=\sqrt{(\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(0.97)/(√(15))=3.25`

`3.79+2.14(0.97)/(√(15))=4.33`

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