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SEVENTH EDITION and EXPANDED SEVENTH EDITION

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SEVENTH EDITION and EXPANDED SEVENTH EDITION

Chapter 13

Statistics

13.1

Sampling Techniques

- 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.

- 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.

- 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.

- 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.

- A raffle ticket is drawn by a blindfolded person at a festival to win a grand prize.
- Students at an elementary school 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.

- 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)Cluster

b)Stratified e)Convenience

c)Systematic

13.2

The 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?

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 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%.

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

13.3

Frequency Distributions

0

0

0

0

0

0

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

3

3

3

3

4

4

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

Number of Pets

Frequency

0

6

0

0

0

0

0

0

1

1

1

1

1

1

1

10

1

1

1

1

2

2

2

8

2

2

2

2

2

2

3

4

3

3

3

3

4

4

4

2

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

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

6.8

5.3

9.7

3.8

8.7

0.5

5.9

0.8

5.7

1.3

4.8

9.6

1.5

7.4

0.2

- 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.

First, rearrange the data from lowest to highest.

# of miles from school

Frequency

0.2

0.5

0.8

0 - 2

5

1.3

1.5

3.8

2.1 - 4.1

1

4.2 - 6.2

4

4.8

5.3

5.7

6.3 - 8.3

2

5.9

6.8

7.4

8.4 -10.4

3

8.7

9.6

9.7

15

13.4

Statistical 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.

Aspirin

56

Ibuprofen

104

Acetaminophen

16

Other

24

200

- 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.

Painkiller

Number of Patients

Percent of Total

Measure of Central Angle

Aspirin

56

0.28 360 = 100.8

Ibuprofen

104

0.52 360 = 187.2

Acetaminophen

16

0.08 360 = 28.8

Other

24

0.12 360 = 43.2

Total

200

100%

360

- Determine the measure of the corresponding central angle.

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

# of pets

Frequency

0

6

1

10

2

8

3

4

4

2

- 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.

# of pets

Frequency

0

6

1

10

2

8

3

4

4

2

- 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.

12

18

3

8

12

25

21

3

15

4

17

27

43

21

16

12

26

35

14

9

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

Data

0

33489

12

18

3

8

12

25

21

3

15

4

1

22245678

17

27

43

21

16

2

11567

12

26

35

14

9

3

5

4

3

13.5

Measures of Central Tendency

- 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, .

- 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.

- 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

middle value (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

7 8

- 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 15

There 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 = 7.5

3 4 4 6 9 11 12 15

- 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.

- 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.

- 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) median
- c) mode d) midrange
- e) rank the measures of central tendency from lowest to highest.

- Mean
- Median-rank the data
- 75, 84, 85, 88, 88, 92, 94, 101
- The median is 88.

- Mode-the number that occurs most frequently. The mode is 88.
- Midrange = (L + H)/2
= (75 + 101)/2 = 88

- Rank the measures
88.375, 88, 88, 88

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

- 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.

- 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, Q3is the median of the data greater than Q2.

- The weekly grocery bills for 23 families are as follows. Determine Q1, Q2, and Q3.
170210270270280

33080170240270

22522521531050

751601307481

95172190

- Order the data:
50 75 74 80 81 95130

160170170172190210215

225225240270270270280

310330

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 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.

- 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

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

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

- 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.

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

Find the mean.

Data

Data Mean

(Data Mean)2

217

297

(297)2 = 88,209

280

234

54,756

299

215

46,225

665

151

22,801

684

170

28,900

939

425

180,625

0

421,516

- The standard deviation is $290.35.

13.7

The Normal Curve

Rectangular Distribution

J-shaped distribution

Bimodal

Skewed to right

Skewed to left

Normal

- 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.

- A normal distribution has a mean of 50 and a standard deviation of 5. Find z-scores for the following values.
- a) 55b) 60c) 43
- a)

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

- 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.

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.

4.

- 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.

- 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%.

- 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.

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%

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.

13.8

Linear Correlation and Regression

- 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.

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

- 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.

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

Applicant

Words per Minute

Mistakes

Ellen

24

8

George

67

11

Phillip

53

12

Kendra

41

10

Nancy

34

9

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.

WPM

Mistakes

x

y

x2

y2

xy

24

8

576

64

192

67

11

4489

121

737

53

12

2809

144

636

41

10

1681

100

410

34

9

1156

81

306

x = 219

y = 50

x2 =10,711

y2 = 510

xy = 2,281

- 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.

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

- 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 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.

- Equation:

- 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.

From the previous results, we know that

Now we find the y-intercept, b.

Therefore the line of best fit isy = 0.081x + 6.452

x

y

10

7.262

20

8.072

30

8.882

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