## Lesson 5: Solving Normal Curve Problems

September 6, 2017

Review:

- Standard Normal Calculations with Z-Scores
- Assignment Grading
**Exam 1 on Wed, Sep 13**

Presentation:

Solving Normal Curve Problems

- Find probabilities for Z-Scores with the Z-Table (use this link or back page of textbook)
**Normal Distribution Probability 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 Probabilities**- Draw a picture of the distribution
- Convert given values (a, b) to Z-Scores and locate on the horizontal axis
- Look up corresponding probabilities 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 probability 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 probabilities from the Z-Table
- Shade the area under the curve between the two Z-Scores
- Find the probability for the larger value (further to the right)
- Find the probability for the smaller value (further to the left)
- Subtract the smaller from the larger to find the “Between” probability

**Inverse Normal Curve calculations**- Z = (x – xbar)/s
- x = xbar + Z * s
- xbar = mean
- s = standard deviation
- Examples
- Step through examples 1.29, 1.30 and 1.31 on pp. 65-66
- What proportion of observations on a standard Normal curve are less than Z=1.47?
- Use 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?

Assignment:

- 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).**- What proportion of students scored less than 600?
- What proportion of students scored greater than 600?
- What proportion of students scored between 600 and 650?
- What score is necessary to be in the top 5% of student test takers?
- 60% of students will score above x on the exam. What is x?

- 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.,
- Problem 2
- Repeat each question in Problem 1 using a different normal distribution of scores:
**N(505,110)**.- What proportion of students scored less than 600?
- What proportion of students scored greater than 600?
- What proportion of students scored between 600 and 650?
- What score is necessary to be in the top 5% of student test takers?
- 60% of students will score above x on the exam. What is x?

- Repeat each question in Problem 1 using a different normal distribution of scores:

Study:

- Read pp. 64-68