Normal Distribution...contribution pls?
The time taken for Mimi to walk from home to school according to a normal distribution is with a mean of 16 minutes and a standard deviation of 3 minutes. She departed from her home at 0700 and the school bell will ring at 0720. Determine the probability that she would be late to school.
The time taken for Mimi to walk from home to school according to a normal distribution is with a mean of 16 minutes and a standard deviation of 3 minutes. She departed from her home at 0700 and the school bell will ring at 0720. Determine the probability that she would be late to school.
1 Answer
The probability is about 0.0918, or 9.18%.
Explanation:
Let
Since the bell rings 20 minutes after Mimi starts walking to school, we want to know the probability that her walk takes longer than 20 minutes:
#"P"(X > 20)#
It would be awesome if we could just look up this probability in a table, right? Well, we don't have a table for every normal distribution—there are just too many of them. But we do have a table for the standard normal distribution
#z = (x-mu)/sigma#
Here,
#P(X < x) = "P"(Z < z)#
That is,
#P(X < x) = "P"(Z < (x-mu)/sigma)#
#P(X < 20) = "P"(Z < (20-16)/3)#
#color(white)(P(X < 20)) = "P"(Z < 4/3)#
#color(white)(P(X < 20)) ~~ "P"(Z < 1.33)#
Now we have a value that we can look up in a table. For the above value of
#"P"(Z < 1.33) = 0.9082#
which means
#"P"(X < 20) = 0.9082#
But wait—this isn't our final answer! The table gives us a probability of being less than a certain value, but we want the probability of being greater than it. We remember that the area under the whole curve of any normal distribution is 1, and so the answer we want is found like this:
#"P"(X > 20) = 1 - "P"(X < 20)#
#color(white)("P"(X > 20)) = 1 - 0.9082#
#color(white)("P"(X > 20)) = 0.0918#
The probability that Mimi is late for school is about 9.18%.