$.)$ It is claimed that the average travel time on a newly built highway between city $A$

and city $B$ is less than $180$ minutes.

a$.$ Write down the null and alternative hypotheses.

Assume the following decision rule: If the travel time of a randomly selected vehicle

going from city A to city B is less than a constant K$,$ the null hypothesis would be

rejected in favor of the alternative hypothesis.

b$.$ Select a sensible value for $K.$

c$.$ Assuming normal distribution with standard deviation $30$ minutes, calculate Type I

error probability.

d$.$ Assuming normal distribution with mean $140$ minutes and standard deviation $30$

minutes, calculate Type II error probability.

e$.$ Calculate the probabilities in $($c$)$ and $($d$),$ if the test would be performed by using the

average travel time of $9$ randomly selected vehicles.