1. Section 9.5
The Bell Curve

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

3. 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 = -

4. x-3s x-2s x-1s x x+1s x+2s x+3s
z-score

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

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

7. 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.

8. 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) = = 45.14%

9. 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.

10. 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) = = 76.98%

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

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

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

14. 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 . =

15. 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.
= = 69.15%
= 69th percentile

16. 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.

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

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

19. Homework pp

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

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

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

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

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

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

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

27. ►B. Exercises
Find the percentage of values in each interval.
≤ z ≤ 2.3

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

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

30. ►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?

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

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

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

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

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