Question

Six measurements were made of the magnesium ion concentration (in parts per million, or ppm) in a city's municipal water supply, with the following results. It is reasonable to assume that the population is approximately normal. Construct a 99% confidence interval for the mean magnesium ion concentration.

Answer 1

Answer:

$163.83-4.03\frac{20.094}{\sqrt{6}}=130.77$    

$163.83+4.03\frac{20.094}{\sqrt{6}}=196.89$    

So on this case the 99% confidence interval would be given by (130.77;196.89)    

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: 175 177 175 180 138 138

We can calculate the mean and the deviation from these data with the following formulas:

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

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

$\bar X=163.83$ represent the sample mean for the sample  

$\mu$ population mean (variable of interest)

s=20.093 represent the sample standard deviation

n=6 represent the sample size  

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:

$df=n-1=6-1=5$

Since the Confidence is 0.99 or 99%, the value of $\alpha=0.01$ and $\alpha/2 =0.005$, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,5)".And we see that $t_{\alpha/2}=4.03$

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

$163.83-4.03\frac{20.094}{\sqrt{6}}=130.77$    

$163.83+4.03\frac{20.094}{\sqrt{6}}=196.89$    

So on this case the 99% confidence interval would be given by (130.77;196.89)