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:

  Distribution Plot Normal, Mean=0, StDev=1 0.0 0.02275 0.1 0.2 0.3 0.4 -2 0 X Density  

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:

  Distribution Plot Normal, Mean=0, StDev=1 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.9452 0.08 0.09 Density 65 73 X  

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 > View Probability
  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.

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