You are the head of a new construction team that wishes to make modifications to an elevator. Before beginning modifications, you must determine some reasonable guesses for the number of times the elevator travels each day.

$($a$)$  So far, you have found that the elevator travels an average of  1 0  times per day. You guess that the probability that the elevator will travel  1 5  or more times per day is at most  0 . 5 .  Is your guess justified? Be sure to show mathematically how you draw your conclusion .

$($b$)$ In your next round of inspections, you find that the number of times the elevator travels per day has a variance of  5 .  You refine your guess and decide that the probability that the elevator will travel  1 5  or more times per day is at most  0 . 1 .  Is your guess justified? Be sure to show mathematically how you draw your conclusion.

$($c$)$  Now your construction team has taken measurements over several weeks and observes that the number of times the elevator travels per day is symmetric about the mean of  1 0 .  With this new information, can you justify your guess from part  ( b )   ( that the probability that the elevator will travel  1 5  or more times per day is at most  0 . 1 ) ?  Be sure to show mat hematically how you draw your conclusion.

$($d$)$ After some final research, you conclude that the number of times that the elevator travels per day is normally distributed with the mean and variance given above. You narrow your guess at the probability that the elevator travels  1 5  or more times per day to at most  0 . 0 2 .  Is your latest guess justified? Be sure to show mathematically how you draw your conclusion