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 18

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 18

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