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.
- On a PC: from the menu select STATISTICS > Distribution Plot
On a Mac: from the menu select Statistics > Probability Distributions > Distribution Plot - Select Display Probability (Note: The default is the standard normal distribution)
- Select A specified X value
- Select Left tail
- 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.
- On a PC: from the menu select STATISTICS > Distribution Plot
On a Mac: from the menu select Statistics > Probability Distributions > Distribution Plot - Select Display Probability
- For Distribution select Normal (Note: This is the default)
- For Mean enter 65
- For Standard deviation enter 5
- Select A specified X value
- Select Left tail
- 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
Select your operating system below to see a step-by-step guide for this example.