Lesson #9: Applications of a Normal Distribution Accel Precalculus Unit #1: Data Analysis Lesson #9: Applications of a Normal Distribution EQ: How is the probability of normal distributions used in the real world?

Recall: Formula for Standardizing Data: z = z =

Find the corresponding z-scores for each raw score: Review Examples. Suppose that the scores for a standardized test are normally distributed with µ = 100 and σ = 10. Sketch both a normal distribution and a standard normal distribution for this data. 70 80 90 100 110 120 130 -3 -2 -1 0 1 2 3 WORK SMARTER, NOT HARDER!!! USE LISTS TO CALCULATE Z-SCORE! Find the corresponding z-scores for each raw score: Raw score L1 z-score formula L2 raw score 120 90 70 110 85 93 113 z-score 2 -1 -3 1 -1.5 -0.7 1.3

Find the percentage of test scores: a) between 100 and 120 P(100 < x < 120) = P(0 < z < 2) = 47.7% b) between 90 and 120 P(90 < x < 120) = P(-1 < z < 2) = 81.9% c) above 93 P(x > 93) = P(z > -.7) = 75.8% e) below 70 d) between 105 and 113 P(105 < x < 113) = P(0.5 < z < 1.3) = 21.2% P(x < 70) = P(z < -3) = 0.13%

For examples #2 – 6, answer the question and sketch a SND for each (mark the z-score and shade). 2. The mean number of hours an American worker spends on the computer is 3.1 hours per workday. Assume the standard deviation is 0.5 hour. Find the percentage of workers who spend less than 3.5 hours on the computer. Assume the variable is normally distributed. P(x < 3.5) = P(z < 0.8) = 78.8%

ASSIGNMENT: Finish the examples in the notes #2 – 6. DG7 --- Fri

P(27 < x < 31) = P(-0.5 < z < 1.5) = 62.5% 3. Each month an American household generates an average of 28 pounds of newspaper for garbage or recycling. Assume the standard deviation of its generating is 2 pounds. If a household is selected at random, find the probability it generates each of the following. Assume the variable is normally distributed. a. between 27 and 31 pounds per month P(27 < x < 31) = P(-0.5 < z < 1.5) = 62.5%

3. Each month an American household generates an average of 28 pounds of newspaper for garbage or recycling. Assume the standard deviation of its generating is 2 pounds. If a household is selected at random, find the probability it generates each of the following. Assume the variable is normally distributed. b. more than 30.2 pound per month P( x > 30.2) = P( z > 1.1) = 13.6%

P( x < 15) = P( z < -2.22) = 1.3% 4. The American Automobile Association (AAA) reports that the average time it takes to respond to an emergency call is 25 minutes. Assume the variable is approximately normally distributed and the standard deviation is 4.5 minutes. If 80 calls are randomly selected, approximately how many will be responded to in less than 15 minutes? P( x < 15) = P( z < -2.22) = 1.3% Approximately only 1 call out of 80 will be responded to in less than 15 minutes. (0.013)(80) = 1.04

z-score from probability! For a medical study, a researcher wished to select people in the middle 60% of the population based on blood pressure. If the mean systolic blood pressure is 120 and the standard deviation is 8, find the upper and lower readings that would qualify people to participate in the study. Assume the blood pressure measurements are normally distributed. 0.84 -0.84 To qualify for this study, the systolic blood pressure reading must be between 113.3 and 126.7. Use invnorm to get z-score from probability!

Use invnorm to get z-score from probability! 6. To qualify for a police academy, candidates must score in the top 10% on a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find the lowest possible score a candidate must have to qualify. Assume the test scores are normally distributed. 90% 10% 1.282 Candidates must make at least 225.63 to qualify for the police academy. Use invnorm to get z-score from probability!

Assignment: PW Applications of the Normal Distribution DG7 --- Fri