Section 9.5 The Bell Curve

Objectives: 1. To use normal curve tables. 2. To find the percentile rank for a score.

It is given by the function The bell-shaped curve is a mound-shaped frequency distribution called the normal distribution. It is given by the function z2 2 1 e 2 y = -

x-3s x-2s x-1s x x+1s x+2s x+3s -3 -2 -1 0 1 2 3 z-score

EXAMPLE 1 Find the percentage of values in the interval 0  z  1.63. 44.84%

Practice: Find the percentage of values in the interval 0  z  1.82. 46.56%

EXAMPLE 2 Find the percentage of values lying within 0 EXAMPLE 2 Find the percentage of values lying within 0.6 standard deviations of the mean. Given the symmetry of the curve you need to find the percent of values in the interval 0  z  0.6 and double it to find -0.6 ≤ z ≤ 0.6.

EXAMPLE 2 Find the percentage of values lying within 0 EXAMPLE 2 Find the percentage of values lying within 0.6 standard deviations of the mean. 0.2257 2(0.2257) = 0.4514 = 45.14%

Practice: Find the percentage of values lying within 1 Practice: Find the percentage of values lying within 1.2 standard deviations of the mean. Given the symmetry of the curve you need to find the percent of values in the interval 0  z  1.2 and double it to find -1.2 ≤ z ≤ 1.2.

Practice: Find the percentage of values lying within 1 Practice: Find the percentage of values lying within 1.2 standard deviations of the mean. 0.3849 2(0.3849) = 0.7698 = 76.98%

EXAMPLE 3 Find the percentage of values such that z  0.98. 0.3365 0.5 - 0.3365 = 0.1635 = 16.35%

Practice: Find the percentage of values such that z  0.8. 0.2881 0.5 - 0.2881 = 0.2119 = 21.19%

Definition Percentile Rank The percentage of values less than or equal to a given value.

EXAMPLE 4 Find the percentile rank of a student whose quiz score is 29 in a class with a mean of 27 and a standard deviation of 4. x s z - = 4 27 29 - = 5 . =

EXAMPLE 4 Find the percentile rank of a student whose quiz score is 29 in a class with a mean of 27 and a standard deviation of 4. 0.5 + 0.1915 = 0.6915 = 69.15% = 69th percentile

Practice: Find the percentile rank of a student whose quiz score is 24 in a class with a mean of 27 and a standard deviation of 4.

EXAMPLE 5 Find the interval of z-scores around the mean that contains 44% of the scores. 0.2190  0.22  0.2224 0.22 is closer to 0.2190 z = 0.58 [-0.58, 0.58]

Practice: Find the interval of z-scores around the mean that contains 52% of the scores. 0.2580  0.26  0.2611 0.26 is closer to 0.2611 z = 0.71 [-0.71, 0.71]

Homework pp. 472-473

►A. Exercises Find the percentage of values in each interval. 1. 0 ≤ z ≤ 1.7

►A. Exercises Find the percentage of values in each interval. 3. -1.21 ≤ z ≤ 0

►A. Exercises Find the percentage of values in each interval. 5. -0.74 ≤ z ≤ 0.74

►A. Exercises Find the percentage of values in each interval. 7. z ≥ 1.06

►A. Exercises Find the percentage of values in each interval. 9. z ≤ -1.56

►A. Exercises Find the percentage of values in each interval. 13. z ≤ 2.0

►B. Exercises Find the interval of z-scores around the mean that contain the following percentage of values. 17. 39%

►B. Exercises Find the percentage of values in each interval. 21. -1.4 ≤ z ≤ 2.3

►B. Exercises Find the percentile rank of each student. 25. Mary’s z-score is -0.26.

►B. Exercises Find the percentile rank of each student. 27. Mark’s score was 14 in a class with a mean of 31 and a standard deviation of 9.

►C. Exercises 29. Bryan had a percentile rank of 91 on a test having a mean of 77 and a standard deviation of 8. What was his score on the original test?

■ Cumulative Review Given a triangle with sides of 17, 29, and 40, find 32. its area.

■ Cumulative Review Given a triangle with sides of 17, 29, and 40, find 33. the measures of its angles to the nearest degree.

■ Cumulative Review Graph the following in polar coordinates. 34. r = cos 

■ Cumulative Review Graph the following in polar coordinates. 35. r = sin 2

■ Cumulative Review Graph the following in polar coordinates. 36. r = 2 – 2 cos 