Answer :
Answer:
95% confidence interval = [ 33.78 , 38.22 ]
Step-by-step explanation:
We are given that average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of 6 minutes.
A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes.
The Pivotal quantity for 95% confidence interval is given by;
[tex]\frac{Xbar - \mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, X bar = sample mean = 36
[tex]\sigma[/tex] = population standard deviation = 6
n = sample size = 28
So, 95% confidence interval for population mean, [tex]\mu[/tex] is given by;
P(-1.96 < N(0,1) < 1.96) = 0.95
P(-1.96 < [tex]\frac{Xbar - \mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < 1.96) = 0.95
P(-1.96 * [tex]{\frac{\sigma}{\sqrt{n} }[/tex] < [tex]{Xbar - \mu}[/tex] < 1.96 * [tex]{\frac{\sigma}{\sqrt{n} }[/tex] ) = 0.95
P(X bar - 1.96 * [tex]{\frac{\sigma}{\sqrt{n} }[/tex] < [tex]\mu[/tex] < X bar + 1.96 * [tex]{\frac{\sigma}{\sqrt{n} }[/tex] ) = 0.95
95% confidence interval for [tex]\mu[/tex] = [ X bar - 1.96 * [tex]{\frac{\sigma}{\sqrt{n} }[/tex] , X bar + 1.96 * [tex]{\frac{\sigma}{\sqrt{n} }[/tex] ]
= [ 36 - 1.96 * [tex]{\frac{6}{\sqrt{28} }[/tex] , 36 + 1.96 * [tex]{\frac{6}{\sqrt{28} }[/tex] ]
= [ 33.78 , 38.22 ]
Therefore, 95% confidence interval for population mean is [33.78 , 38.22] .