Joe Straphanger takes MARTA every day to get to work. His commute time on MARTA

Question

Joe Straphanger takes MARTA every day to get to work. His commute time on MARTA each day is normally distributed with a mean of 16.71 minutes and a standard deviation of 2 minutes independent of the time it takes on any other day. If Joe’s commute time on MARTA is greater than 20 minutes he will be late to work. a) What is the probability Joe will be late on any given day?
Joe plans on recording his commute time for the next 100 days and he will calculate the average of these commute times.
.b) What is the distribution of Joe’s average commute time?
.c) What is the probability that his average commute time is less than 17 minutes?
.d) Let X be the number of days out of the next 100 that Joe is late. What is the distribution of X?
.e) Write down an expression for the probability that Joe is late 6 of the next 100 days he rides MARTA. You do not have to calculate the expression but make sure it is as complete as possible. Joe Straphanger takes MARTA every day to get to work. His commute time on MARTA each day is normally distributed with a mean of 16.71 minutes and a standard deviation of 2 minutes independent of the time it takes on any other day. If Joe’s commute time on MARTA is greater than 20 minutes he will be late to work. a) What is the probability Joe will be late on any given day?
Joe plans on recording his commute time for the next 100 days and he will calculate the average of these commute times.
.b) What is the distribution of Joe’s average commute time?
.c) What is the probability that his average commute time is less than 17 minutes?
.d) Let X be the number of days out of the next 100 that Joe is late. What is the distribution of X?
.e) Write down an expression for the probability that Joe is late 6 of the next 100 days he rides MARTA. You do not have to calculate the expression but make sure it is as complete as possible. Joe Straphanger takes MARTA every day to get to work. His commute time on MARTA each day is normally distributed with a mean of 16.71 minutes and a standard deviation of 2 minutes independent of the time it takes on any other day. If Joe’s commute time on MARTA is greater than 20 minutes he will be late to work. a) What is the probability Joe will be late on any given day?
Joe plans on recording his commute time for the next 100 days and he will calculate the average of these commute times.
.b) What is the distribution of Joe’s average commute time?
.c) What is the probability that his average commute time is less than 17 minutes?
.d) Let X be the number of days out of the next 100 that Joe is late. What is the distribution of X?
.e) Write down an expression for the probability that Joe is late 6 of the next 100 days he rides MARTA. You do not have to calculate the expression but make sure it is as complete as possible.

Explanation / Answer

Here mean commute time = 16.71 minutes

Standard deviation of commute time = 2 minutes

(a) Here if X is the comute time of Joe on a random day.

Pr(X > 20 minutes) = NORM(X > 20 minutes ; 16.71 mins ; 2 mins) = 1 -NORM(X < 20 mins, 16.71 mis, 2 mis)

Z = (20 - 16.71)/2 = 1.645

So the p -value from Z table is 0.95

Pr(X > 20 minutes) = 1 - 0.95 = 0.05

so 5% probability that he will be late.

(b) Here now n = 100 days

Average commute time have the normal ditribution with mean value of 16.71 minutes and standard deviation = 2/sqrt(100) = 2/10= 0.2 minutes

(c) Pr(x < 17 minutes ; 16.71 ; 0.2) where x is the average commute time

Z = (17 - 16.71)/ 0.2 = 1.45

Pr(x < 17 minutes ; 16.71 ; 0.2) = Pr(Z < 1.45) = 0.9265

(d) Here the given distribution of X would be binomial distrbituon with paramenter n = 100 and p = 0.05

(e) Pr(X = 6 ) = BIN(X = 6; 100 ; 0.05) = 100C6 (0.05)6(0.95)94

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