Question

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.

Answer 1

The probability is about 0.0918, or 9.18%.

Explanation:

Let `X` be how long it takes Mimi to walk to school. Then `X` is distribued as `"Norm"(mu=16, sigma=3).`

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`, and a way to convert a point on any normal distribution (like our `X`) to its corresponding point on `Z`. The conversion is done like this:

`z = (x-mu)/sigma`

Here, `z` is the point on the standard normal distribution that matches up with the point `x` on a normal distribution that has mean `mu` and standard deviation `sigma.` What do we mean by "matches up"? We mean this:

`P(X < x) = "P"(Z < z)`

That is, `x` and `z` both mark a point on their respective distributions with the same amount of area to their left (and also to their right). And since `z = (x-mu)/sigma`, we plug in the value of `x=20` (along with `mu = 16` and `sigma = 3`) to get

`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 `z=1.33`, our table tells us that

`"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%.