A realtor would like to create a 95% confidence interval for the average price of a house in Franklin County, Ohio. To accomplish this, she collected a random sample of 529 houses and calculated a sample mean price of $320,128. She knows that the population standard deviation for house prices in Franklin County is $50,324. What is her 95% confidence interval?

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".

[tex]\bar X=320128[/tex] represent the sample mean

[tex]\mu[/tex] population mean (variable of interest)

[tex]\sigma=50324[/tex] represent the population standard deviation

n=529 represent the sample size

Solution to the problem

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

[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)

Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that [tex]z_{\alpha/2}=1.96[/tex]

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

[tex]320128-1.96\frac{50324}{\sqrt{529}}=315839.52[/tex]

[tex]320128+1.96\frac{50324}{\sqrt{529}}=324416.48[/tex]

So on this case the 95% confidence interval would be given by (315839.52;324416.48)