Use the given degree of confidence and sample data to construct a confidence interval for the population mean mu. Assume that the population has a normal distribution. n=30, x=84.6, s=10.5, 90% confidence A. 81.36B. 80.68C. 79.32D. 81.34

Answer:Option A - The population mean is [tex]\mu\approx 81.36[/tex]Confidence interval is (3.153,166.047).Step-by-step explanation:Given : Assume that the population has a normal distribution n=30, x=84.6, s=10.5, 90% confidence.To find : Use the given degree of confidence and sample data to construct a confidence interval for the population mean mu?Solution : First we apply z-score formula,[tex]z=\frac{x-\mu}{\frac{s}{\sqrt{n}}}[/tex]Where, x is the sample mean x=84.6s is the standard deviation s=10.5n is the number of sample n=30[tex]\mu[/tex] is the population meanz is the score value, at 90% z=1.645Substitute all the values in the formula,[tex]1.645=\frac{84.6-\mu}{\frac{10.5}{\sqrt{30}}}[/tex][tex]1.645\times \frac{10.5}{\sqrt{30}}=84.6-\mu[/tex]    [tex]1.645\times 1.917=84.6-\mu[/tex]    [tex]3.153=84.6-\mu[/tex]    [tex]\mu=84.6-3.153[/tex]    [tex]\mu=81.447[/tex]    So, The population mean is [tex]\mu=81.447[/tex]Therefore, Option A is correct.Now, Apply confidence interval formula[tex]x-\mu<CI<x+\mu[/tex][tex]84.6-81.447<CI<84.6+81.447[/tex][tex]3.153<CI<84.6+166.047[/tex]Therefore, Confidence interval is (3.153,166.047).