## Lesson 5: Normal Curve and Z-Scores

September 9, 2019

Review:

- Quiz 2
- Standard Deviation
**Exam 1 on Wed Sep 18**

Presentation:

- Normal Curves
- Density Curves
- Height of curve indicates proportion of values
- Area under the curve = 1.0
- Any sub-area under the curve is then a proportion (% of values)

- Normal Curves
- A special case of density curves
- Bell shaped and symmetrical
- Mean and Standard deviation
- 68 – 95 – 99.7 Rule
- Video (until 4:30)
- Demonstrate with student height data

- Density Curves
- Standard Normal Distribution
- Mean = 0
- Standard Deviation = 1
- Calculating Z-Scores
- (x – mean)/std dev
- Example (p. 61)
- Young women heights normally distributed
- Mean = 64.5 in
- Standard Deviation = 2.5 in
- Z-Score for woman 68 inches tall
- Z = (68 – 64.5)/2.5 = 1.4

- Z-Score for woman 60 inches tall
- Z = (60 – 64.5)/2.5 = -1.8

- Calculating proportions/probabilities with Z-scores

Activity:

**Problem 1.** The distribution of heights of young women (18 – 24 years old) is approximately normal with mean = 64.5 inches and standard deviation = 2.5 inches.

- Draw a normal curve with a horizontal axis. Label mean.
- What is the z-score for a 20-yr-old woman who is 6 ft tall? Label point.
- Between what two values do the heights of the central 95% of young women lie? Shade this area.
- What % of young women are more than two standard deviations taller than the mean?

**Problem 2.** There are two national college-entrance examinations, the SAT and the ACT. Scores on individual SAT exams are approximately normal with mean = 500 and standard deviation=100. Scores on the ACT exams are approximately normal with mean = 18 and standard deviation = 6.

- Julie’s SAT Math score is 630. John’s ACT Math score is 22. Calculate the standardized Z-scores for both.
- Assuming that both tests measure the same kind of ability, who has the higher score?
- What percent of all SAT scores are above 600?

**Problem 3.** Find the percentage of observations from a standard normal distribution that satisfy the following. Also, draw a normal curve and shade the corresponding area under the curve.

- z < −1
- z < 1
- z > -2
- z > 2
- −1 < z < 2
- -2 < z < 0

Assignment:

- Text Ch. 1 pp. 61-64