# 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?

Answer:$$320128-1.96\frac{50324}{\sqrt{529}}=315839.52$$    $$320128+1.96\frac{50324}{\sqrt{529}}=324416.48$$    So on this case the 95% confidence interval would be given by (315839.52;324416.48)    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". $$\bar X=320128$$ represent the sample mean $$\mu$$ population mean (variable of interest) $$\sigma=50324$$ 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: $$\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$$   (1) 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: "=-NORM.INV(0.025,0,1)".And we see that $$z_{\alpha/2}=1.96$$ Now we have everything in order to replace into formula (1): $$320128-1.96\frac{50324}{\sqrt{529}}=315839.52$$    $$320128+1.96\frac{50324}{\sqrt{529}}=324416.48$$    So on this case the 95% confidence interval would be given by (315839.52;324416.48)