7.2 - Minitab Express: Finding Proportions

7.2 - Minitab Express: Finding Proportions

Minitab Express can be used to find the proportion of a normal distribution in a given range. The default is to construct a standard normal distribution (i.e., z distribution), but the mean and standard deviation of the distribution can be edited. The following pages walk through how to construct normal distributions to find the proportion greater than a given value, the proportion less than a given value, or the proportion between two given values.

Later in this lesson, we'll see that these procedures may be used to find the p value for a given test statistic. For a right-tailed test, the p value is the area greater than the test statistic. For a left-tailed test the p value is the area less than the test statistic. For a two-tailed test, the p value is the total area in the left and right tails that is more extreme than the test statistic. 


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:

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

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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

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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


7.2.2 - Proportion 'Greater Than'

7.2.2 - Proportion 'Greater Than'

MinitabExpress  – Proportion Greater Than a z Value

Question: What proportion of the standard normal distribution is greater 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 Right tail
  5. For X value enter 2

This should result in the following output:

Minitab Express output: z distribution showing the area above 2

The area of the z distribution that is greater than 2 is 0.0227501

Video Walkthrough

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

MinitabExpress  – Proportion Greater Than a Value on a Normal Distribution

Question: 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 more than 73 mph? 

Let's construct a normal distribution with a mean of 65 and standard deviation of 5 to find the area greater than 73.

To calculate a probability for values greater than a given value 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 
  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 Right tail
  8. For X value enter 73

This should result in the following output:

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

On a normal distribution with a mean of 65 and standard deviation of 5, the proportion greater than 73 is 0.0547993

In other words, 5.47993% of vehicles will be going more than 73 mph.

Video Walkthrough

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7.2.2.1 - Video Example: P(Z>0.5)

7.2.2.1 - Video Example: P(Z>0.5)

Question: What proportion of the z distribution is greater than z = 0.5?

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 Right tail
  5. For X value enter 0.5
Video Walkthrough

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7.2.3 - Proportion 'In between'

7.2.3 - Proportion 'In between'

MinitabExpress  – Proportion Between Two z Values

Question: What proportion of the standard normal distribution is between a z score of 0 and a z score of 1.75?

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 Middle
  5. For X value 1 enter 0 and for X value 2 enter 1.75

This should result in the following output:

Standard normal distribution from Minitab Express showing that the area between 0 and 1.75 is 0.459941

The proportion of the z distribution that is between 0 and 1.75 is 0.459941

Video Walkthrough

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MinitabExpress  – Proportion Between Values on a Normal Distirbution

Question: 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 between 60 mph and 73 mph?

Let's construct a normal distribution with a mean of 65 and standard deviation of 5 to find the area between 60 and 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
  4. For Mean enter 65
  5. For Standard deviation enter 5
  6. Select A specified X value
  7. Select Middle
  8. For X value 1 enter 60 and for X value 2 enter 73

This should result in the following output:

Distribution Plot - Normal, Mean=65, StDev=5; In Between

On a normal distribution with a mean of 65 and standard deviation of 5, the proportion between 60 and 73 is 0.786545

In other words, 78.6545% of vehicles will be going between 60 mph and 73 mph. 

Video Walkthrough

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7.2.3.1 - Video Example: Proportion Between z -2 and +2

7.2.3.1 - Video Example: Proportion Between z -2 and +2

Question: What proportion of the z distribution is between -2 and 2?

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 Middle
  5. For X value 1 enter -2 and for X value 2 enter 2

7.2.4 - Proportion 'More Extreme Than'

7.2.4 - Proportion 'More Extreme Than'

MinitabExpress

Question: What proportion of the standard normal distribution is more extreme than a z value of ±2?

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 Equal tails
  5. For X value enter 2

This should result in the following output:

Minitab Express output showing a z distribution showing the area less than -2 and greater than +2

Video Walkthrough

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