# 7.2.1 - Proportion 'Less Than'

7.2.1 - Proportion 'Less Than'The **cumulative probability** for a value is the probability less than or equal to that value. In notation, this is \(P(X\leq x)\). The proportion at or below a given value is also known as a **percentile**.

##
Minitab^{®}
– Proportion Less Than a z Value

**Question**: What proportion of the standard normal distribution is less than a z score of -2?

Recall that the standard normal distribution (i.e., z distribution) has a mean of 0 and standard deviation of 1. This is the default normal distribution in Minitab.

- From the tool bar select
*Graph > Probability Distribution Plot > One Curve > View Probability* - Check that the
*Mean*is 0 and the*Standard deviation*is 1 - Select
*Options* - Select
*A specified x value* - Select
*Left tail* - For
*X value*enter -2 - Click
*Ok* - Click
*Ok*

This should result in the following output:

The proportion of the standard normal distribution that is less than a z score of -2 is 0.02275.

This could also be written as P(z < -2) = 0.02275.

##
Minitab^{®}
– Proportion Less Than a Value on a Normal Distribution

**Scenario: **Vehicle speeds at a highway location have a normal distribution with a mean of 65 mph and a standard deviation of 5 mph. What is the probability that a randomly selected vehicle will be going 73 mph or slower?

Let's construct a normal distribution with a mean of 65 and standard deviation of 5 to find the area less than 73.

- From the tool bar select
*Graph > Probability Distribution Plot > One Curve > View Probability* - Change the
*Mean*to 65 and the*Standard deviation*to 5 - Select
*Options* - Select
*A specified x value* - Select
*Left tail* - For
*X value*enter 73 - Click
*Ok* - Click
*Ok*

This should result in the following output:

On a normal distribution with a mean of 65 mph and standard deviation of 5 mph, the proportion less than 73 mph is 0.9452.

In other words, 94.52% of vehicles will be going 73 mph or slower.

# 7.2.1.1 - Example: P(Z<-1)

7.2.1.1 - Example: P(Z<-1)**Question:** What proportion of the z distribution falls below a z score of -1?

- In Minitab select
*Graph > Probability Distribution Plot > One Curve > View Probability*, hit*OK*. - Select
*Normal*(Note: The default is the standard normal distribution) - Select
*Options* - Select
*A specified x value* - Select
*Left Tail* - For
*X value*enter -1 - OK

# 7.2.1.2 - Example: P(SATM<540)

7.2.1.2 - Example: P(SATM<540)**Question:** SAT-Math scores are normally distributed with a mean of 500 and standard deviation of 100. What proportion of scores are less than 540?

- In Minitab choose
*Graph > Probability Distribution Plot* - For
*Distribution*select*Normal*(Note: This is the default) - For
*Mean*enter 500 - For
*Standard deviation* - Select
*Options* - Select
*A specified X value* - Select
*Left tail* - For
*X value*enter 540