Do Now: 1. Scott was one of 50 junior boys to take the PSAT at his school. He scored 64 on the Critical Reading test. This placed Scott at the 68th percentile within the group of boys. Looking at all 50 boys Critical Reading scores, the mean was 58.2 and the standard deviation was 9.4. Calculate and compare Scott s z-score among the group of test takers 2. Scores on the ACT college entrance exam follow a bell-shaped distribution with mean 18 and standard deviation 6. Wayne s standardized score on the ACT was 0.7. What was Wayne s actual ACT score? (a) 4.2 (b) 4.2 (c) 13.8 (d) 17.3 (e) 22.2 Normal Distributions All Normal curves are symmetric, single-peaked, and bell-shaped A Specific Normal curve is described by giving its mean µ and standard deviation σ. The mean is located at the center of the symmetric curve and is the same as the median. The standard deviation σ controls the spread of a Normal curve. Curves with larger standard deviations are more spread out.
The 68-95-99.7 Rule Although there are many Normal curves, they all have properties in common. In the Normal distribution with mean μ and standard deviation σ: Approximately 68% of the observations fall within 1σ of the mean μ. Approximately 95% of the observations fall within 2σ of μ. Approximately 99.7% of the observations fall within 3σ of μ
Using the 68 95 99.7 rule Example 1: Suppose we know that a distribution is exactly Normal for the scores of a New York State 6th grade vocabulary exam with mean μ = 6.84 and standard deviation σ = 1.55. (a) Sketch a Normal curve for this distribution of test scores. Label the points that are one, two, and three standard deviations from the mean. (b) What percent of the NYS vocabulary scores are less than 3.74? Show your work. (c) What percent of the scores are between 5.29 and 9.94? Show your work. Example 2: The distribution of heights of young women aged 18 to 24 is a normal distribution with a mean µ = 64.5 and σ = 2.5. (a) Sketch a Normal curve for the distribution of young women s heights. Label the points one, two, and three standard deviations from the mean. (b) What percent of young women have heights greater than 67 inches? (c) What percent of young women have heights between 62 and 72 inches?
Example 3: Suppose we know that the average (μ) high school relationship is 42 days with a standard deviation (σ) of 2 days. (a) Assuming that the distribution of days is approximately Normal, make an accurate sketch of the distribution with the horizontal axis marked in days. (b) Between which standard deviations does the interval from 38-42 lie? (c) What percentage represents this interval? (d) What does this percentage represent in the context of this problem?
LESSON PRACTICE 1. The distribution of heights of adult American men is approximately Normal with mean 69 inches and standard deviation 2.5 inches. Draw a Normal curve on which this mean and standard deviation are correctly located. (a) What percent of men are taller than 74 inches? (b) Between what heights do the middle 95% of men fall? (c) What percent of men are between 64 and 66.5 inches tall? 2. The distribution of weights of 9-ounce bags of a particular brand of potato chips is approximately Normal with mean μ = 9.12 ounces and standard deviation σ = 0.05 ounce. Draw an accurate sketch of the distribution of potato chip bag weights. (a) What percent of bags weigh less than 9.02 ounces? (b) Between what weights do the middle 68% of bags fall? (c) What percent of 9-ounce bags of this brand of potato chips weigh between 8.97 and 9.17 ounces?
LESSON PRACTICE 3. The average (μ) attention span for teenagers is 10 minutes, with a standard deviation (σ) of 1 min. Draw an accurate sketch of this distribution. (a) Between which standard deviations does the interval from 7-11 lie? (b) What percentage represents this interval? (c) What does this percentage represent in the context of this problem? (d) Between which standard deviations does the interval from 11-13 lie? (e) What percentage represents this interval? (f) What does this percentage represent in the context of this problem?