A simple random sample of 125 SAT scores has a mean of 1522. Assume that SAT scores have a population standard deviation of 335. Construct a 95% confidence interval estimate of the mean SAT score and then correctly interpret your interval.

Accepted Solution

Answer:[tex]1463.3\:<\:\mu\:<\:1580.7[/tex]Step-by-step explanation:Since the population standard deviation [tex]\sigma[/tex], is known, we use the z confidence interval for the mean.This is given by:[tex]\bar X-z_{\frac{\alpha}{2} }(\frac{\sigma}{\sqrt{n} } )\:<\:\mu\:<\bar X+z_{\frac{\alpha}{2} }(\frac{\sigma}{\sqrt{n} } )[/tex]For a 95% confidence interval we use [tex]z=1.96[/tex].It was also given that: [tex]n=125[/tex], [tex]\bar X=1522[/tex] and [tex]\sigma=335[/tex]Let us substitute the values to get:[tex]1522-1.96(\frac{335}{\sqrt{125} } )\:<\:\mu\:<\:1522+1.96(\frac{335}{\sqrt{125} } )[/tex][tex]1463.3\:<\:\mu\:<\:1580.7[/tex]Interpretation:We can say with 95% confidence that the interval between 1463.3 and 1580.7 SAT scores contains the population mean based on the sample 125 SAT scores.