7.3 - Minitab: Finding Values Given Proportions

7.3 - Minitab: Finding Values Given Proportions

Minitab can also be used to find the values that separate a given proportion of the normal distribution. This can be used to find the value that offset a given proportion, such as the top 10%, bottom 25%, or middle 95%. In this lesson, we'll learn how to find such values on the z distribution or on a normal distribution with a given mean and standard deviation.

In Lesson 4, we used the standard error method to construct a 95% confidence interval by estimating the z* multiplier to be 2 using the Empirical Rule, because approximately 95% of a normal distribution falls within two standard deviations of the mean. Later in this lesson, we'll see that the procedures we're learning here, specifically finding the z scores that offset the middle X%, can be used to determine the z* multiplier to construct a confidence interval for any confidence level. For example, we can use Minitab to find the z values that offset the middle 90% of the z distribution, which would be the multipliers for a 90% confidence interval.


7.3.1 - Top X%

7.3.1 - Top X%

On this page, we'll focus on finding the values that offset the top X% of a normal distribution, for example the top 10% or top 20%. The first example below uses the standard normal distribution. The second exam uses a normal distribution with a mean of 85 and standard deviation of 5.

Minitab®  – z Score Separating the Top X%

Question: What z score separates the top 10% of the z distribution from the bottom 90%?

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 probability
  5. Select Right tail
  6. For Probability enter 0.10
  7. Click Ok
  8. Click Ok

This should result in the following output:

Standard normal distribution constructed with Minitab showing the top 10%

A z score of 1.282 separates the top 10% of the z distribution from the bottom 90%.

 

Video Walkthrough

Minitab®  – Value Separating the Top X%

Question: Scores on a test are normally distributed with a mean of 85 points and standard deviation of 5 points. What score separates the top 10% from the bottom 90%?

Steps
  1. From the tool bar select Graph > Probability Distribution Plot > One Curve > View Probability
  2. Change the Mean to 85 and the Standard deviation to 5
  3. Select Options
  4. Select A specified probability
  5. Select Right tail
  6. For Probability enter 0.10
  7. Click Ok
  8. Click Ok

This should result in the following output:

Normal distribution constructed in Minitab with a mean of 85 and standard deviation of 5 showing the top 10%

The test score that separates the top 10% from the bottom 90% is 91.41 points. This could also be described as the 90th percentile.

 

Video Walkthrough


7.3.2 - Bottom X%

7.3.2 - Bottom X%

Next, we'll find the z scores or observations that off set the bottom X% of a normal distribution. Earlier in this lesson, we learned that this is also known as the cumulative proportion or percentile. The first example below uses the z distribution. The second example uses a normal distribution with a mean of 85 and standard deviation of 5.

Minitab®  – z Score Separating the Bottom X%

Question: What z score separates the bottom 10% of the standard normal distribution from the top 90%?

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 probability
  5. Select Left tail
  6. For Probability enter 0.10
  7. Click Ok
  8. Click Ok

This should result in the following output:

z distribution constructed with Minitab showing the bottom 10%

A z score of -1.282 separates the bottom 10% of the z distribution from the top 90%.

Video Walkthrough

Minitab®  – Value on a Normal Distribution Separating the Bottom X%

Question: Scores on a test are normally distributed with a mean of 85 points and standard deviation of 5 points. What score is the 10th percentile? In other words, what score separates the bottom 10% from the top 90% of this distribution?

Steps
  1. From the tool bar select Graph > Probability Distribution Plot > One Curve > View Probability
  2. Change the Mean to 85 and the Standard deviation to 5
  3. Select Options
  4. Select A specified probability
  5. Select Left tail
  6. For Probability enter 0.10
  7. Click Ok
  8. Click Ok

This should result in the following output:

Normal distribution constructed in Minitab with a mean of 85 and standard deviation of 5 showing the bottom 10%

The 10th percentile on this test is a score of 78.59 points.

Video Walkthrough


7.3.3 - Middle X%

7.3.3 - Middle X%

Here, we'll use Minitab to find the points on a normal distribution that offset the most extreme X%. The first example below uses the z distribution, which later in the lesson we'll see can be plugged into the formula for a confidence interval to obtain an interval with any confidence level. For example, the z scores that separate the middle 90% from the outer 10% could be used to compute a 90% confidence interval. The second example below is similar, but it uses a distribution with a mean of 85 and standard deviation of 5.

Note that in Minitab, the proportion you will enter is the total proportion in the two tails combined. Minitab will split that proportion equally between the left and right tails.

Minitab®  – z Scores Separating the Middle X%

Question: What z scores separate the middle 90% of the z distribution from the most extreme 10%?

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 probability
  5. Select Equal tails
  6. For Probability enter 0.10
  7. Click Ok
  8. Click Ok

This should result in the following output:

z distribution constructed with Minitab showing the most extreme 10%

The z scores of ±1.645 separate the middle 90% of the z distribution from the outer 10% .

Video Walkthrough

Minitab®  – Values on a Normal Distribution Separating the Middle X%

Question: Scores on a test are normally distributed with a mean of 85 points and standard deviation of 5 points. What scores separate the middle 90% from the most extreme 10%?

Steps
  1. From the tool bar select Graph > Probability Distribution Plot > One Curve > View Probability
  2. Change the Mean to 85 and the Standard deviation to 5
  3. Select Options
  4. Select A specified probability
  5. Select Equal tails
  6. For Probability enter 0.10
  7. Click Ok
  8. Click Ok

This should result in the following output:

Normal distribution constructed in Minitab with a mean of 85 and standard deviation of 5 showing the most extreme 10%

The middle 90% of scores are between 76.78 points and 93.22 points.

Video Walkthrough


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