Lesson 6: Solving Normal Curve Problems

September 11, 2019

Review:

• Review Quiz 2 and Averages
• Standard Normal Calculations with Z-Scores
• Exam 1 on Wed, Sep 18

Presentation:

Solving Normal Curve Problems

• Video (0:45-3:45, 6:10-11:45)
• Find probabilities for Z-Scores with the Z-Table (use this link or back page of textbook)
• Normal Distribution Problems: 3 Types
  • p(X<a) = “Less Than”
  • p(X>b) = “Greater Than”
  • p(a<X<b) = “Between” (between 2 values)
• Steps to finding Standard Normal Proportions
  • Draw a picture of the distribution
  • Convert given values (a, b) to Z-Scores and locate on the horizontal axis
  • Look up corresponding proportion in the Z-Table
  • Decide if it’s a “Less Than”, “Greater Than” or “Between” problem
    • If “Less Than”, shade under the curve to the left of Z-Score; the Z-Table proportion you found is the answer.
    • If “Greater Than”, shade under the curve to the right of the Z-Score; subtract the Z-Table probability from 1 to find the answer.
    • If “Between”, you will have 2 proportions from the Z-Table
      • Shade the area under the curve between the two Z-Scores
      • Find the proportion for the larger value (further to the right)
      • Find the proportion for the smaller value (further to the left)
      • Subtract the smaller from the larger to find the “Between” proportion
• Inverse Normal Curve calculations
  • Z = (x – xbar)/s
  • x = xbar + Z * s
  • xbar = mean
  • s = standard deviation

• Problem Examples (see pp. 63-67)
  • What proportion of observations on a standard Normal curve are less than Z=1.47?
  • Using N(1026,209) …
    • What proportion of students who take the SAT have scores of at least 820?
    • What proportion of students who take the SAT would be NCAA “partial qualifiers”, i.e., between 720 and 820.
    • What score is necessary to place in the top 10% of all students taking the SAT?

Activity:

• Problem 1
  • In a recent year, 10th grade students took a standardized English language exam. The mean score was 572 and the standard deviation was 51, i.e., N(572,51).
    1. What proportion of students scored less than 600?
    2. What proportion of students scored greater than 600?
    3. What proportion of students scored between 600 and 650?
    4. What score is necessary to be in the top 5% of student test takers?
    5. 60% of students will score above x on the exam. What is x?
• Problem 2
  • Repeat each question in Problem 1 using a different normal distribution of scores: N(505,110).
    1. What proportion of students scored less than 600?
    2. What proportion of students scored greater than 600?
    3. What proportion of students scored between 600 and 650?
    4. What score is necessary to be in the top 5% of student test takers?
    5. 60% of students will score above x on the exam. What is x?

Study:

• Read Text pp. 62-68