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:

Minitab Express output: z distribution showing the proportion less than -2

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:

Distribution Plot - Normal, Mean=65, StDev=5; Less Than

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.