1. Section 6.3 Finding Probability Using the Normal Curve HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

2. HAWKES LEARNING SYSTEMS math courseware specialists Four basic types of probability problems: 1.Probability less than some value 2.Probability greater than some value 3.Probability between two values 4.Probability less than one value and greater than another value Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve

3. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: Deviation IQ scores, sometimes called Wechsler IQ scores, are scores with a mean of 100 and a standard deviation of 15. What percentage of the general population have IQ’s lower than 92? Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   100,   15, x  92 P(z <  0.53)  0.2981  29.81%

4. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: Deviation IQ scores, sometimes called Wechsler IQ scores, are scores with a mean of 100 and a standard deviation of 15. What percentage of the general population have IQ’s larger than 130? Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   100,   15, x  130 P(z > 2.00)  0.0228  2.28%

5. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: Deviation IQ scores, sometimes called Wechsler IQ scores, are scores with a mean of 100 and a standard deviation of 15. What percentage of the general population have IQ’s between 90 and 110? Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   100,   15, x 1  90 and x 2  110 P(  0.67 < z < 0.67)  0.4972  49.72%

6. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: Deviation IQ scores, sometimes called Wechsler IQ scores, are scores with a mean of 100 and a standard deviation of 15. What percentage of the general population have IQ’s less than 80 and greater than 120? Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   100,   15, x 1  80 and x 2  120 P(z 1.33)  0.1835  18.35%

7. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: In a recent year, the ACT scores for high school students with a 3.50 to 4.00 GPA were normally distributed, with a mean of 24.2 and a standard deviation of 4.2. A student who took the ACT during this time is selected. Find the probability that the student’s ACT score is less than 20. Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   24.2,   4.2, x  20 P(z <  1.00)  0.1587  15.87%

8. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: In a recent year, the ACT scores for high school students with a 3.50 to 4.00 GPA were normally distributed, with a mean of 24.2 and a standard deviation of 4.2. A student who took the ACT during this time is selected. Find the probability that the student’s ACT score is greater than 31. Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   24.2,   4.2, x  31 P(z > 1.62)  0.0526  5.26%

9. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: In a recent year, the ACT scores for high school students with a 3.50 to 4.00 GPA were normally distributed, with a mean of 24.2 and a standard deviation of 4.2. A student who took the ACT during this time is selected. Find the probability that the student’s ACT score is between 25 and 32. Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   24.2,   4.2, x 1  25 and x 2  32 P(0.19 < z < 1.86)  0.3933  39.33%

10. Repeating examples with TI-84 shortcuts (added content by D.R.S.) OLD – we’ve done this for z problems: normalcdf(low z, high z) = area between z’s NEW – SHORTCUT FOR x problems: normalcdf(low x, high x, mean μ, stdev σ) = area between x’s on the normal distribution with the given mean and standard deviation This saves you the trouble of changing the x problem into a z problem.

11. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: Deviation IQ scores, sometimes called Wechsler IQ scores, are scores with a mean of 100 and a standard deviation of 15. What percentage of the general population have IQ’s lower than 92? Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   100,   15, x  92 P(x <  )  0.2981  29.81% 29.69% normalcdf(-1E99, 92, 100, 15)=.2969014046

12. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: Deviation IQ scores, sometimes called Wechsler IQ scores, are scores with a mean of 100 and a standard deviation of 15. What percentage of the general population have IQ’s larger than 130? Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   100,   15, x  130 P(x > 130)  0.0228  2.28% normalcdf(130,1E99,100, 15)=.022750062

13. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: Deviation IQ scores, sometimes called Wechsler IQ scores, are scores with a mean of 100 and a standard deviation of 15. What percentage of the general population have IQ’s between 90 and 110? Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   100,   15, x 1  90 and x 2  110 P(90 < x < 110)  0.4972  49.72% 49.50% normalcdf(90, 110, 100, 15)=.495015066

14. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: Deviation IQ scores, sometimes called Wechsler IQ scores, are scores with a mean of 100 and a standard deviation of 15. What percentage of the general population have IQ’s less than 80 and greater than 120? Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   100,   15, x 1  80 and x 2  120 P(x 120)  0.1835  18.35% 18.24% normalcdf(-1E99, 80, 100, 15)+ normalcdf(120,1E99,100, 15)=.1824225637

15. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: In a recent year, the ACT scores for high school students with a 3.50 to 4.00 GPA were normally distributed, with a mean of 24.2 and a standard deviation of 4.2. A student who took the ACT during this time is selected. Find the probability that the student’s ACT score is less than 20. Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   24.2,   4.2, x  20 P(x <  20)  0.1587  15.87% normalcdf(-1E99, 20, 24.2, 4.2)=.1586552596

16. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: In a recent year, the ACT scores for high school students with a 3.50 to 4.00 GPA were normally distributed, with a mean of 24.2 and a standard deviation of 4.2. A student who took the ACT during this time is selected. Find the probability that the student’s ACT score is greater than 31. Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   24.2,   4.2, x  31 P(x > 31)  0.0526  5.26% 5.27% normalcdf(31, 1E99, 24.2, 4.2)=.0527185009

17. HAWKES LEARNING SYSTEMS math courseware specialists Determine the probability: In a recent year, the ACT scores for high school students with a 3.50 to 4.00 GPA were normally distributed, with a mean of 24.2 and a standard deviation of 4.2. A student who took the ACT during this time is selected. Find the probability that the student’s ACT score is between 25 and 32. Continuous Random Variables 6.3 Finding the Probability Using the Normal Curve Solution:   24.2,   4.2, x 1  25 and x 2  32 P(25 < x < 32)  0.3933  39.33% 39.28% normalcdf(25, 32, 24.2, 4.2)=.3928226602