# 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.

Steps
1. From the tool bar select Graph > Probability Distribution Plot > One Curve > View Probability
2. Check that the Mean is 0 and the Standard deviation is 1
3. Select Options
4. Select A specified x value
5. Select Left tail
6. For X value enter -2
7. Click Ok
8. 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.

Video Walkthrough

## 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.

Steps
1. From the tool bar select Graph > Probability Distribution Plot > One Curve > View Probability
2. Change the Mean to 65 and the Standard deviation to 5
3. Select Options
4. Select A specified x value
5. Select Left tail
6. For X value enter 73
7. Click Ok
8. 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.

Video Walkthrough

# 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?

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

The proportion of the z distributions that falls below -1 is 0.1587.

# 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?

Steps
1. In Minitab choose Graph > Probability Distribution Plot
2. For Distribution select Normal (Note: This is the default)
3. For Mean enter 500
4. For Standard deviation enter 100
5. Select Options
6. Select A specified X value
7. Select Left tail
8. For X value enter 540

The proportion of scores less than 540 is 0.6554.

 [1] Link ↥ Has Tooltip/Popover Toggleable Visibility