Lesson 5: Normal Curve and Z-Scores

September 3, 2024

Review:

• Standard Deviation
• Exam 1 on Thu Sep 12

Presentation:

• Normal Curves
  • Normal Curve explained in 1 min
  • Normal Curves explained on 1 page
  • 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 Empirical Rule
    • Against All Odds – Video Unit 7
    • Demonstrate with student height data
• Standard Normal Distribution
  • Mean = 0
  • Standard Deviation = 1
  • Calculating Z-Scores
    • (x – mean)/std dev
    • Example (see text 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

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