copyright seventh edition and expanded seventh edition

Copyright © 2005 Pearson Education, Inc.

Chapter 13

Statistics


13.1

Sampling Techniques

4Copyright © 2005 Pearson Education, Inc.

Statistics

Statistics is the art and science of gathering, analyzing, and making inferences from numerical information (data) obtained in an experiment.Statistics are divided into two main braches.

Descriptive statistics is concerned with the collection, organization, and analysis of data.Inferential statistics is concerned with the making of generalizations or predictions of the data collected.


Statisticians

A statistician’s interest lies in drawing conclusions about possible outcomes through observations of only a few particular events.

The population consists of all items or people of interest. The sample includes some of the items in the population.

When a statistician draws a conclusion from a sample, there is always the possibility that the conclusion is incorrect.


Types of Sampling

A random sampling occurs if a sample is drawn in such a way that each time an item is selected, each item has an equal chance of being drawn.When a sample is obtained by drawing every nth item on a list or production line, the sample is a systematic sample.A cluster sample is referred to as an area sample because it is applied on a geographical basis.


Types of Sampling continued

Stratified sampling involves dividing the population by characteristics such as gender, race, religion, or income.Convenience sampling uses data that is easily obtained and can be extremely biased.


Example: Identifying Sampling Techniques

A raffle ticket is drawn by a blindfolded person at a festival to win a grand prize.Students at an elementary are classified according to their present grade level. Then, a random sample of three students from each grade are chosen to represent their class.Every sixth car on highway is stopped for a vehicle inspection.


Example: Identifying Sampling Techniques continued

Voters are classified based on their polling location. A random sample of four polling locations are selected. All the voters from the precinct are included in the sample.The first 20 people entering a water park are asked if they are wearing sunscreen.

Solution:a) Random d) Clusterb) Stratified e) Conveniencec) Systematic


13.2

The Misuses of Statistics


Misuses of Statistics

Many individuals, businesses, and advertising firms misuse statistics to their own advantage.When examining statistical information consider the following:

Was the sample used to gather the statistical data unbiased and of sufficient size?Is the statistical statement ambiguous, could it be interpreted in more than one way?


Example: Misleading Statistics

An advertisement says, “Fly Speedway Airlines and Save 20%”.

Here there is not enough information given. The “Save 20%” could be off the original ticket price, the ticket price when you buy two tickets or of another airline’s ticket price.

A helped wanted ad read,”Salesperson wanted for Ryan’s Furniture Store. Average Salary: $32,000.”

The word “average” can be very misleading. If most of the salespeople earn $20,000 to $25,000 and the owner earns $76,000, this “average salary” is not a fair representation.


Charts and Graphs

Charts and graphs can also be misleading. Even though the data is displayed correctly, adjusting the vertical scale of a graph can give a different impression.A circle graph can be misleading if the sum of the parts of the graphs do not add up to 100%.


Example: Misleading Graphs

Sales

0255075

100125150175

99 00 01 02 03 04

Years

Dol

lars

(in

thou

sand

s)

Sales

100

200

300

400

500

99 00 01 02 03 04

Years

Dol

lars

(in

thou

sand

s)

While each graph presents identical information, the vertical scales have been altered.


13.3

Frequency Distributions


Example

The number of pets per family is recorded for 30 families surveyed. Construct a frequency distribution of the following data:

443333

2

2

1

0

22222

21111

11111

00000


Solution

24

43

82

101

60

FrequencyNumber of Pets

443333

2

2

1

0

22222

21111

11111

00000


Rules for Data Grouped by Classes

The classes should be of the same “width.”The classes should not overlap.Each piece of data should belong to only one class.


Definitions

Midpoint of a class is found by adding the lower and upper class limits and dividing the sum by 2.

Classes 0 4 5 910 14

Lower class limits Upper class limits15 1920 2425 29

−⎧ ⎫⎪ ⎪−⎪ ⎪⎪ ⎪−⎨ ⎬−⎪ ⎪⎪ ⎪−⎪ ⎪

−⎩ ⎭


Example

The following set of data represents the distance, in miles, 15 randomly selected second grade students live from school.

Construct a frequency distribution with the first class 0 − 2.

0.27.41.59.64.8

1.35.70.85.90.5

8.73.89.75.36.8


Solution

First, rearrange the data from lowest to highest.

9.79.68.7

7.46.85.9

5.75.34.8

3.81.51.3

0.80.50.2

15

38.4 -10.4

26.3 - 8.3

44.2 - 6.2

12.1 - 4.1

50 - 2

Frequency# of miles from school


13.4

Statistical Graphs


Circle Graphs

Circle graphs (also known as pie charts) are often used to compare parts of one or more components of the whole to the whole.


Example

According to a recent hospital survey of 200 patients the following table indicates how often hospitals used four different kinds of painkillers. Use the information to construct a circle graph illustrating the percent each painkiller was used.

200

24Other

16Acetaminophen

104Ibuprofen

56Aspirin


Solution

Determine the measure of the corresponding central angle.

200

24

16

104

56

Number of Patients

100%

Percent of Total

360°

0.12 × 360 = 43.2°

0.08 × 360 = 28.8°

0.52 × 360 = 187.2°

0.28 × 360 = 100.8°

Measure of Central Angle

Total

Other

Acetaminophen

Ibuprofen

Aspirin

Painkiller

56200 100 28%× =

104200 100 52%× =

16200 100 8%× =

24200 100 12%× =


Solution continued

Use a protractor to construct a circle graph and label it properly.

Hospital Painkiller Use

Aspirin28%

Ibuprofen52%

AcetaminopheOther


Histogram

A histogram is a graph with observed values on its horizontal scale and frequencies on it vertical scale.Example: Construct a histogram of the frequency distribution.

24

43

82

101

60

Frequency# of pets


Solution

Number of Pets per Family

0

2

4

6

8

10

12

0 1 2 3 4

Number of Pets

Freq

uenc

y

24

43

82

101

60

Frequency# of pets


Frequency Polygon

Number of Pets per Family

0

2

4

6

8

10

12

0 1 2 3 4

Number of Pets

Freq

uenc

y


Stem-and-Leaf Display

A stem-and-leaf display is a tool that organizes and groups the data while allowing us to see the actual values that make up the data.The left group of digits is called the stem.The right group of digits is called the leaf.


Example

The table below indicates the number of miles 20 workers have to drive to work. construct a stem-and-leaf display.

914352612

1621432717

41532125

12831812


Solution

Data

914352612

1621432717

41532125

12831812

34

53

115672

222456781

334890


13.5

Measures of Central Tendency


Definitions

An average is a number that is representative of a group of data.

The arithmetic mean, or simply the mean is symbolized by or by the Greek letter mu, µ.x


Mean

The mean, is the sum of the data divided by the number of pieces of data. The formula for calculating the mean is

where represents the sum of all the data and n represents the number of pieces of data.

x

xx

n= ∑

x∑


Example-find the mean

Find the mean amount of money parents spent on new school supplies and clothes if 5 parents randomly surveyed replied as follows: $327 $465 $672 $150 $230

$327 $465 $672 $150 $2305

$1844 $368.805

xx

n+ + + +

= =

= =

∑


Median

The median is the value in the middle of a set of ranked data.Example: Determine the mean of $327 $465 $672 $150 $230.Rank the data from smallest to largest.$150 $230 $327 $465 $672

middle value (median)


Example: Median (even data)

Determine the median of the following set of data: 8, 15, 9, 3, 4, 7, 11, 12, 6, 4.Rank the data:3 4 4 6 7 8 9 11 12 15There are 10 pieces of data so the median will lie halfway between the two middle pieces the 7 and 8. The median is (7 + 8)/2 = 7.53 4 4 6 9 11 12 157 8


Mode

The mode is the piece of data that occurs most frequently.

Example: Determine the mode of the data set: 3, 4, 4, 6, 7, 8, 9, 11, 12, 15.The mode is 4 since is occurs twice and the other values only occur once.


Midrange

The midrange is the value halfway between the lowest (L) and highest (H) values in a set of data.

Example: Find the midrange of the data set $327, $465, $672, $150, $230.

lowest value + highest valueMidrange = 2

$150 + $672Midrange = $4112

=


Example

The weights of eight Labrador retrievers rounded to the nearest pound are 85, 92, 88, 75, 94, 88, 84, and 101. Determine the

a) mean b) medianc) mode d) midrangee) rank the measures of central tendency

from lowest to highest.


Example--dog weights 85, 92, 88, 75, 94, 88, 84, 101

Mean

Median-rank the data75, 84, 85, 88, 88, 92, 94, 101The median is 88.

85 92 88 75 94 88 84 1018

707 88.3758

x + + + + + + +=

= =


Example--dog weights 85, 92, 88, 75, 94, 88, 84, 101

Mode-the number that occurs most frequently. The mode is 88.Midrange = (L + H)/2

= (75 + 101)/2 = 88Rank the measures

88.375, 88, 88, 88


Measures of Position

Measures of position are often used to make comparisons.Two measures of position are percentiles and quartiles.


To Find the Quartiles of a Set of Data

Order the data from smallest to largest.Find the median, or 2nd quartile, of the set of data. If there are an odd number of pieces of data, the median is the middle value. If there are an even number of pieces of data, the median will be halfway between the two middle pieces of data.


To Find the Quartiles of a Set of Data continued

The first quartile, Q1, is the median of the lower half of the data; that is, Q1, is the median of the data less than Q2.The third quartile, Q3, is the median of the upper half of the data; that is, Q3 is the median of the data greater than Q2.


Example: Quartiles

The weekly grocery bills for 23 families are as follows. Determine Q1, Q2, and Q3.

170 210 270 270 280330 80 170 240 270225 225 215 310 5075 160 130 74 8195 172 190


Example: Quartiles continued

Order the data:50 75 74 80 81 95 130

160 170 170 172 190 210 215225 225 240 270 270 270 280310 330

Q2 is the median of the entire data set which is 190.Q1 is the median of the numbers from 50 to 172 which is 95.Q3 is the median of the numbers from 210 to 330 which is 270.


13.6

Measures of Dispersion


Measures of Dispersion

Measures of dispersion are used to indicate the spread of the data.

The range is the difference between the highest and lowest values; it indicates the total spread of the data.


Example: Range

Nine different employees were selected and the amount of their salary was recorded. Find the range of the salaries.

$24,000 $32,000 $26,500$56,000 $48,000 $27,000$28,500 $34,500 $56,750

Range = $56,750 − $24,000 = $32,750


Standard Deviation

The standard deviation measures how much the data differ from the mean.

( )2

1

x xs

n

−=

−∑


To Find the Standard Deviation of a Set of Data

1. Find the mean of the set of data.2. Make a chart having three columns:

Data Data − Mean (Data − Mean)2

3. List the data vertically under the column marked Data.

4. Subtract the mean from each piece of data and place the difference in the Data − Mean column.


To Find the Standard Deviation of a Set of Data continued

5. Square the values obtained in the Data −Mean column and record these values in the(Data − Mean)2 column.

6. Determine the sum of the values in the (Data − Mean)2 column.

7. Divide the sum obtained in step 6 by n − 1, where n is the number of pieces of data.

8. Determine the square root of the number obtained in step 7. This number is the standard deviation of the set of data.


Example

Find the standard deviation of the following prices of selected washing machines: $280, $217, $665, $684, $939, $299

Find the mean.

665 217 684 280 939 299 3084 5146 6

xx

n+ + + + +

= = = =∑


Example continued, mean = 514

421,5160

180,625425939

28,900170684

22,801151665

46,225−215299

54,756−234280

(−297)2 = 88,209−297217

(Data − Mean)2Data − MeanData


Example continued, mean = 514

The standard deviation is $290.35.

421,5166 1

421,516 290.355

s

s

=−

= =


13.7

The Normal Curve


Types of Distributions

Rectangular Distribution J-shaped distribution

Rectangular Distribution

Values

Freq

uenc

y


Types of Distributions continued

Bimodal Skewed to right


Types of Distributions continued

Skewed to left Normal


Normal Distribution

In a normal distribution, the mean, median, and mode all have the same value.Z-scores determine how far, in terms of standard deviations, a given score is from the mean of the distribution.

value of piece of data meanstandard deviation

xz µσ

− −= =


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)

55

value of piece of data meanstandard deviation

55 50 55

15

z

z

−=

−= = =

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


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.

6060 50 10 2

5 5z −

= = =

4343 50 7 1.4

5 5z − −

= = = −


To Find the Percent of Data Between any Two Values1. 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.


To Find the Percent of Data Between any Two Values continued4.

a) When finding the percent of data between two z-scores on the opposite side of the mean (when one z-score is positive and the other is negative), you find the sum of the individual percents.b) When finding the percent of data between two z-scores on the same side of the mean (when both z-scores are positive or both are negative), subtract the smaller percent from the larger percent.


To Find the Percent of Data Between any Two Values continued

c) When finding the percent of data to the right of a positive z-score or to the left of a negative z-score, subtract the percent of data between ) and z from 50%.d) When finding the percent of data to the left of a positive z-score or to the right of a negative z-score, add the percent of data between 0 and z to 50%.


Example

Assume that the waiting times for customers at a popular restaurant before being seated for lunch 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.


Solution

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.

between 9 and 18 minutes

Use table 13.7 page 801.34.1% + 47.7%

= 81.8%

9

18

9 12 1.003

18 12 2.003

z

z

−= = −

−= =


Solution continued

at least 17 min

Use table 13.7 page 801.45.3% is between the mean and 1.67. 50% − 45.3% = 4.7%Thus, 4.7% of customers wait at least 17 minutes.

less than 8 min

Use table 13.7 page 801.40.8% is between the mean and −1.33. 50% − 40.8% = 9.2%Thus, 9.2% of customers wait less than 8 minutes.

1717 12 1.67

3z −

= = 88 12 1.33

3z −

= = −


13.8

Linear Correlation and Regression


Linear Correlation

Linear correlation is used to determine whether there is a relationship between two quantities and, if so, how strong the relationship is.

The linear correlation coefficient, r, is a unitless measure that describes the strength of the linear relationship between two variables.

If the value is positive, as one variable increases, the other increases.If the value is negative, as one variable increases, the other decreases.The variable, r, will always be a value between –1 and 1 inclusive.


Scatter Diagrams

A visual aid used with correlation is the scatter diagram, a plot of points (bivariate data).

The independent variable, x, generally is a quantity that can be controlled.The dependant variable, y, is the other variable.

The value of r is a measure of how far a set of points varies from a straight line.

The greater the spread, the weaker the correlation and the closer the r value is to 0.


Correlation


Linear Correlation Coefficient

The formula to calculate the correlation coefficient (r) is as follows:

( ) ( )( )( ) ( ) ( ) ( )2 22 2

n xy x yr

n x x n y y

∑ − ∑ ∑=

∑ − ∑ ∑ − ∑


There are five applicants applying for a job as a medical transcriptionist. The following shows the results of the applicants when asked to type a chart. Determine the correlation coefficient between the words per minute typed and the number of mistakes.

Example: Words Per Minute versus Mistakes

934Nancy

1041Kendra

1253Phillip

1167George

824Ellen

MistakesWords per MinuteApplicant


We will call the words typed per minute, x, and the mistakes, y.List the values of x and y and calculate the necessary sums.

Solution

306811156934

xy = 2,281y2 = 510x2 =10,711y = 50x = 219

10

12

11

8

y

Mistakes

xyy2x2x

41

53

67

24

WPM

4101001681

6361442809

7371214489

19264576

∑∑ ∑ ∑∑


Solution continued

The n in the formula represents the number of pieces of data. Here n = 5.

( ) ( )( )( ) ( ) ( ) ( )

( ) ( )( )( ) ( ) ( ) ( )

( ) ( )

2 22 2

2 2

5 2281 219 50

5 10,711 219 5 510 50

11,405 10,9505 10,711 47,961 5 510 2500

45553,555 47,961 2550 2500

455 0.865594 50

n xy x yr

n x x n y y

r

∑ − ∑ ∑=

∑ − ∑ ∑ − ∑

−=

− −

−=

− −

=− −

= ≈


Solution continued

Since 0.86 is fairly close to 1, there is a fairly strong positive correlation.This result implies that the more words typed per minute, the more mistakes made.


Linear Regression

Linear regression is the process of determining the linear relationship between two variables. The line of best fit (line of regression or the least square line) is the line such that the sum of the vertical distances from the line to the data points is a minimum.


The Line of Best Fit

Equation:

( ) ( )( )( ) ( )

( )22

, where

, and

y mx b

n xy x y y m xm b

nn x x

= +

∑ − ∑ ∑ ∑ − ∑= =

∑ − ∑


Example

Use the data in the previous example to find the equation of the line that relates the number of words per minute and the number of mistakes made while typing a chart.Graph the equation of the line of best fit on a scatter diagram that illustrates the set of bivariate points.


Solution

From the previous results, we know that

Now we find the y-intercept, b.

( ) ( )( )( ) ( )22

2

5(2,281) (219)(50)5(10,711) 219

45555940.081

n xy x ym

n x x

m

m

m

∑ − ∑ ∑=

∑ − ∑

−=

−

=

≈

( )

( )50 0.081 2195

32.261 6.4525

y m xb

n

b

b

∑ − ∑=

−=

= ≈

Therefore the line of best fit is y = 0.081x + 6.452


Solution continued

To graph y = 0.081x + 6.452, plot at least two points and draw the graph.

8.88230

8.07220

7.26210

yx


Solution continued

copyright seventh edition and expanded seventh edition

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