# 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.

## MinitabExpress – 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., distribution) has a mean of 0 and standard deviation of 1. This is the default normal distribution in Minitab Express.

Steps
1. On a PC: from the menu select STATISTICS > Distribution Plot
On a Mac: from the menu select Statistics > Probability Distributions > Distribution Plot
2. Select Display Probability (Note: The default is the standard normal distribution)
3. Select A specified X value
4. Select Left tail
5. For X value enter -2

This should result in the following output:

The proportion of the z distribution that is less than -2 is 0.0227501.

#### Video Walkthrough

Select your operating system below to see a step-by-step guide for this example.

## MinitabExpress – 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. On a PC: from the menu select STATISTICS > Distribution Plot
On a Mac: from the menu select Statistics > Probability Distributions > Distribution Plot
2. Select Display Probability
3. For Distribution select Normal (Note: This is the default)
4. For Mean enter 65
5. For Standard deviation enter 5
6. Select A specified X value
7. Select Left tail
8. For X value enter 73

This should result in the following output:

On a normal distribution with a mean of 65 and standard deviation of 5, the proportion less than 73 is 0.945201

In other words, 94.5201% of vehicles will be going less than 73 mph.

#### Video Walkthrough

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

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

Question: What proportion of the z distribution falls below a z score of -1?

Steps
1. On a PC: from the menu select STATISTICS > Distribution Plot

On a Mac: from the menu select Statistics > Probability Distributions > Distribution Plot

2. Select Display Probability (Note: The default is the standard normal distribution)
3. Select A specified X value
4. Select Left tail
5. For X value enter -1
Video Walkthrough

This video is available in both Windows and Mac platforms. Select your platform below.

# 7.2.1.2 - Video Example: P(SATM<540)

7.2.1.2 - Video 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. On a PC: from the menu select STATISTICS > Distribution Plot
On a Mac: from the menu select Statistics > Probability Distributions > Distribution Plot
2. Select Display Probability
3. For Distribution select Normal (Note: This is the default)
4. For Mean enter 500
5. For Standard deviation enter 100
6. Select A specified X value
7. Select Left tail
8. For X value enter 540

#### Video Walkthrough

 [1] Link ↥ Has Tooltip/Popover Toggleable Visibility