Chapter 13 Section 7 – Slide 1Copyright © 2009 Pearson Education, Inc.

AND

Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 2

Chapter 13

Statistics

Chapter 13 Section 7 – Slide 3Copyright © 2009 Pearson Education, Inc.

WHAT YOU WILL LEARN• Sampling techniques• Misuses of statistics• Frequency distributions• Histograms, frequency polygons,

stem-and-leaf displays• Mode, median, mean, and

midrange• Percentiles and quartiles

Chapter 13 Section 7 – Slide 4Copyright © 2009 Pearson Education, Inc.

WHAT YOU WILL LEARN

• Range and standard deviation• z-scores and the normal distribution• Correlation and regression

Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 5

Section 7

The Normal Curve

Chapter 13 Section 7 – Slide 6Copyright © 2009 Pearson Education, Inc.

Types of Distributions

Rectangular Distribution J-shaped distribution

Rectangular Distribution

Values

Fre

quen

cy

Chapter 13 Section 7 – Slide 7Copyright © 2009 Pearson Education, Inc.

Types of Distributions (continued)

Bimodal Skewed to right

Chapter 13 Section 7 – Slide 8Copyright © 2009 Pearson Education, Inc.

Types of Distributions (continued)

Skewed to left Normal

Chapter 13 Section 7 – Slide 9Copyright © 2009 Pearson Education, Inc.

Properties of a Normal Distribution

The graph of a normal distribution is called the normal curve.

The normal curve is bell shaped and symmetric about the mean.

In a normal distribution, the mean, median, and mode all have the same value and all occur at the center of the distribution.

Chapter 13 Section 7 – Slide 10Copyright © 2009 Pearson Education, Inc.

Empirical Rule

Approximately 68% of all the data lie within one standard deviation of the mean (in both directions).

Approximately 95% of all the data lie within two standard deviations of the mean (in both directions).

Approximately 99.7% of all the data lie within three standard deviations of the mean (in both directions).

Chapter 13 Section 7 – Slide 11Copyright © 2009 Pearson Education, Inc.

z-Scores

z-scores determine how far, in terms of standard deviations, a given score is from the mean of the distribution.

z

value of piece of data mean

standard deviation

x

Chapter 13 Section 7 – Slide 12Copyright © 2009 Pearson Education, Inc.

Example: z-scores

A normal distribution has a mean of 50 and a standard deviation of 5. Find z-scores for the following values.

a) 55 b) 60 c) 43

a)

A score of 55 is one standard deviation above the mean.

z value of piece of data mean

standard deviation

z55

55 50

5

5

51

Chapter 13 Section 7 – Slide 13Copyright © 2009 Pearson Education, Inc.

Example: z-scores (continued)

b)

A score of 60 is 2 standard deviations above the mean.

c)

A score of 43 is 1.4 standard deviations below the mean.

z

60

60 50

5

10

52

z

43

43 50

5

7

5 1.4

Chapter 13 Section 7 – Slide 14Copyright © 2009 Pearson Education, Inc.

To Find the Percent of Data Between any Two Values

1. Draw a diagram of the normal curve, indicating the area or percent to be determined.

2. Use the formula to convert the given values to z-scores. Indicate these z-scores on the diagram.

3. Look up the percent that corresponds to each z-score in Table 13.7.

Chapter 13 Section 7 – Slide 15Copyright © 2009 Pearson Education, Inc.

To Find the Percent of Data Between any Two Values (continued) 4.

a) When finding the percent of data to the left of a negative z-score, use Table 13.7(a).

b) When finding the percent of data to the left of a positive z-score, use Table 13.7(b).

c) When finding the percent of data to the right of a z-score, subtract the percent of data to the left of that z-score from 100%.

d) When finding the percent of data between two z-scores, subtract the smaller percent from the larger percent.

Chapter 13 Section 7 – Slide 16Copyright © 2009 Pearson Education, Inc.

Example

Assume that the waiting times for customers at a popular restaurant before being seated for lunch are normally distributed with a mean of 12 minutes and a standard deviation of 3 min.

a) Find the percent of customers who wait for at least 12 minutes before being seated.

b) Find the percent of customers who wait between 9 and 18 minutes before being seated.

c) Find the percent of customers who wait at least 17 minutes before being seated.

d) Find the percent of customers who wait less than 8 minutes before being seated.

Chapter 13 Section 7 – Slide 17Copyright © 2009 Pearson Education, Inc.

Solution

a. wait for at least 12 minutes Since 12 minutes is the mean, half, or 50% of customers wait at least 12 min before being seated.

b. between 9 and 18 minutes

Use table 13.7 on pages 889-89 in the 8th edition.97.7% - 15.9% = 81.8%

z9

9 12

3 1.00

z18

18 12

32.00

Chapter 13 Section 7 – Slide 18Copyright © 2009 Pearson Education, Inc.

Solution (continued)

c. at least 17 min

Use table 13.7(b) page 889.100% - 95.3% = 4.7%Thus, 4.7% of customers wait at least 17 minutes.

d. less than 8 min

Use table 13.7(a) page 889.Thus, 9.2% of customers wait less than 8 minutes.

z

17

17 12

31.67

z

8

8 12

3 1.33