Question

From 19 measurements of a pressure controller, the mean operating pressure is 4.97 kPa with a standard deviation of 0.0461 kPa. Find: If more measurements are taken, estimate the expected range of pressure for 95% of the measurements.

Answer 1

Answer:

$4.97-2.10\frac{0.0461}{\sqrt{19}}=4.95$    

$4.97+2.10\frac{0.0461}{\sqrt{19}}=4.99$    

So on this case the 95% confidence interval would be given by (4.95;4.99)  

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".

$\bar X =4.97$ represent the sample mean

$\mu$ population mean (variable of interest)

s=0.0461 represent the sample standard deviation

n=19 represent the sample size  

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:

$df=n-1=19-1=18$

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(0.025,18)".And we see that $t_{\alpha/2}=2.10$

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

$4.97-2.10\frac{0.0461}{\sqrt{19}}=4.95$    

$4.97+2.10\frac{0.0461}{\sqrt{19}}=4.99$    

So on this case the 95% confidence interval would be given by (4.95;4.99)