Statistics Chapter 9

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Statistics Chapter 9. Day 1. Unusual Episode. MS133 Final Exam Scores. 79 86 79 65 78 91 78 94 88 75 71 53 95 96 79 62 79 67 64 77 69 58 74 69 78 78 91 89 49 68 63 77 86 84 77. Line Plot or Dot Plot. Stem and Leaf. Stem and Leaf. Ordered Stem and Leaf.

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Statistics Chapter 9

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MS133 Final Exam Scores

79 86 79 65 78

91 78 94 88 75

71 53 95 96 79

62 79 67 64 77

69 58 74 69 78

78 91 89 49 68

63 77 86 84 77

5 A’s out of how many total grades? 35
• What percent of the class made an A?
5 A’s out of how many total grades? 35
• What percent of the class made an A?

5/35 ≈ 0.14 ≈14%

• What percent of the pie should represent the A’s?
5 A’s out of how many total grades? 35
• What percent of the class made an A?

5/35 ≈ 0.14 ≈14%

• What percent of the pie should represent the A’s? 14%
• How many degrees in the whole pie?
5 A’s out of how many total grades? 35
• What percent of the class made an A?

5/35 ≈ 0.14 ≈14%

• What percent of the pie should represent the A’s? 14%
• How many degrees in the whole pie? 360°
5 A’s out of how many total grades? 35
• What percent of the class made an A?

5/35 ≈ 0.14 ≈14%

• What percent of the pie should represent the A’s? 14%
• How many degrees in the whole pie? 360°
• 14% of 360° is how many degrees?
5 A’s out of how many total grades? 35
• What percent of the class made an A?

5/35 ≈ 0.14 ≈14%

• What percent of the pie should represent the A’s? 14%
• How many degrees in the whole pie? 360°
• 14% of 360° is how many degrees?

.14 x 360° ≈ 51°

14 C’s out of 35 grades
• 14/35 = .4 = 40%
• .4 x 360° = 144°
8 D’s out of 35 grades
• 8/35 ≈ .23 ≈ 23% (to the nearest percent)

(keep the entire quotient in the calculator)

• x 360° ≈ 82°
3 F’s out of 35 grades total
• 3/35 ≈ .09 ≈ 9% (to the nearest percent)

(keep the entire quotient in the calculator)

• x 360° ≈ 31°
• Check the remaining angle to make sure it is 31°
Make a Pie Chart
• Gross income: \$10,895,000
• Labor: \$5,120,650
• Materials: \$4,031,150
• New Equipment: \$326,850
• Plant Maintenance: \$544,750
• Profit: \$871,600
Labor: \$5,120,650 = 47% 169°

10,895,000

• Materials: \$4,031,150 = 37% 133°

10,895,000

• New Equipment: \$326,850 = 3% 11°

10,895,000

• Plant Maintenance: \$544,750 = 5% 18°

10,895,000

• Profit : \$871,600 = 8% 29°

10,895,000

Histogram
• Table 9.2 Page 527
Measures of Central Tendency Lab

Getting Mean with Tiles
• Use your colored tiles to build a column 9 tiles high and another column 15 tiles high. Use a different color for each column.
Getting Mean with Tiles
• Use your colored tiles to build a column 9 tiles high and another column 15 tiles high. Use a different color for each column.
• Move the tiles one at a time from one column to another “evening out” to create 2 columns the same height.
• What is the new (average) height?
Getting Mean with Tiles
• Move the tiles back so that you have a column 9 tiles high and another 15 tiles high.
• Find another method to “even off” the columns?
Getting Mean with Tiles
• Use your colored tiles to build a column 19 tiles high and another column 11 tiles high. Use a different color for each column.
• “Even-off” the two columns using the most efficient method.
• What is the new (average) height?
Getting Mean with Tiles
• If we start with a column x tiles high and another y tiles high, describe how you could find the new (average) height?
• Let’s assume x is the larger number
x – y(extra)x – y 2
• y + x – y

2

2y + x – y

2 2

x – y(extra)x – y 2
• y + x – y

2

2y + x – y

2 2

2y + x - y

2

x – y(extra)x – y 2
• y + x – y

2

2y + x – y

2 2

2y + x - y

2

x + y 2

Measures of Central Tendency
• Mean – “Evening-off”
• Median – “Middle”
• Most – “Most”
Class R

71 71 76 79 77 76 70 72 92 74 86 79 46 79 72 81 67 77 72 77 63 77 61 76

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,9246,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Mean = Sum of all grades

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,9246,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Mean = Sum of all grades

Mean = 1771

24

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,9246,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Mean = Sum of all grades

Mean = 1771

24

Class S

72 77 75 75 67 76 69 76 71 68 77 79 82 73 69 76 68 69 71 78 72 79 74 73 73

67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,8267,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Mean =

67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,8267,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Mean =

67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,8267,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Mean =

Class T

74 79 86 84 40 82 40 61 40 49 70 85 49 40 45 91 74 96 81 85 86 75 89 85

40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,9640,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Mean =

40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,9640,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Mean =

40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,9640,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Mean =

Median –”Middle”

Class R:

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Class S:

67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Class T:

40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Median
• Class R: 76
• Class S: 73
• Class T: 77
Mode – “Most”

Class R:

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Class S:

67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Class T:

40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Mode
• Class R: 77
• Class S: 69, 73, 76
• Class T: 40
Range - A measure of dispersionGreatest - Least

Class R:

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Class S:

67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Class T:

40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Range
• Class R: 92 - 46 = 46
• Class S: 82 – 67 = 15
• Class T: 96 – 40 = 56
Class RClass SClass T

Mean = 73.873.6 70.3

Median = 76 73 77

Mode = 77 69,73,76 40

Range = 46 15 56

Weighted Mean Example 9.7

Owner/Manager earned \$850,000

Assistant Manager earned \$48,000

16 employees \$27,000 each

3 secretaries \$18,000 each

Find the MEAN, MEDIAN, MODE

MEAN

Mean = 3(18,000)+16(27,000)+48,000+850,000 21

= 1384000 21

≈ \$65,905

Mean = \$65,905
• Median = \$27,000
• Mode = \$27,000
• Range = \$832,000

quality points earned

hours attempted

Quality Points

Every A gets 4 quality points per hour. For example, an A in a 3 hour class gets 4 quality points for each of the 3 hours, 4x3=12. An A in a 4 hour class gets 4 quality points for each of the 4 hours, 4X4=16 quality points.

Every B gets 3 quality points per hour.

Every C gets 2 quality points per hour.

Every D gets 1 quality points per hour.

No quality points for an F.

Sally Ann’s Cumulative GPA

Total quality points earned

Total hours attempted

Class X

60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90

Find the mean, median, mode, and range.

Mean

60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90

Mean

60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90

Median – Mode – Range

60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90

Mean = 78
• Median = 82
• Mode = 82
• Range = 30
Standard Deviation

The standard deviation is a measure of dispersion. You can think of the standard deviation as the “average” amount each data is away from the mean. Some data are close, some are farther. The standard deviation gives you an average.

Find the standard deviation of class x.

Standard Deviation

60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90

Mean = 78

Page 558Example 9.11

Find the mean (to the nearest tenth):

35, 42, 61, 29, 39

Page 558Example 9.11

Find the mean (to the nearest tenth): ≈41.2

Standard deviation (to the nearest tenth):

35, 42, 61, 29, 39

Page 558Example 9.11

Find the mean (to the nearest tenth): ≈41.2

Standard deviation (to the nearest tenth): ≈ 10.8

Box and Whisker Graph
• Graph of dispersion
• Data is divided into fourths
• The middle half of the data is in the box
• Outliers are not connected to the rest of the data but are indicted by an asterisk.
Box and Whisker Graph
• Class R:

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =

Upper Quartile = Lower Quartile =

Outliers
• Any data more than 1 ½ boxes away from the box (middle half) is considered an outlier and will not be connected to the rest of the data.
• The size of the box is called the Inner Quartile Range (IQR) and is determined by finding the range of the middle half of the data.
Box and Whisker Graph
• Class R:

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range =

Box and Whisker Graph
• Class R:

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range = 7 IQR x 1.5 =

Box and Whisker Graph
• Class R:

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range = 7 IQR x 1.5 = 10.5

Checkpoints for Outliers:

Box and Whisker Graph
• Class R:

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range = 7 IQR x 1.5 = 10.5

Checkpoints for Outliers: 60.5, 88.5

Outliers =

Box and Whisker Graph
• Class R:

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range = 7 IQR x 1.5 = 10.5

Checkpoints for Outliers: 60.5, 88.5

Outliers = 46, 92 Whisker Ends =

Box and Whisker Graph
• Class R:

46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92

Median =76

Upper Quartile = 78 Lower Quartile = 71

Inner Quartile Range = 7 IQR x 1.5 = 10.5

Checkpoints for Outliers: 60.5, 88.5

Outliers = 46, 92 Whisker Ends = 61, 86

Box and Whisker Graph
• Class S:

67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Median =

UQ = LQ =

IQR = IQR x 1.5 =

Checkpoints for outliers:

Outliers = Whisker Ends =

Box and Whisker Graph
• Class S:

67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82

Median = 73

UQ = 76.5 LQ = 70

IQR = 6.5 IQR x 1.5 = 9.75

Checkpoints for outliers: 60.25, 86.25

Outliers = none Whisker Ends = 67, 82

Box and Whisker Graph
• Class T:

40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Median =

UQ = LQ = IQR =

IQR x 1.5 =

Checkpoints for Outliers:

Outliers= Whisker Ends=

Box and Whisker Graph
• Class T:

40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96

Median = 77

UQ = 85 LQ = 49 IQR = 36

IQR x 1.5 = 54

Checkpoints for Outliers: -5, 139

Outliers = none Whisker Ends = 40, 96

Statistical Inference
• Population
• Sampling
• Random Sampling
• Page 576 #2, 4, 5, 17, 18, 19, 21, 22
Example 9.15, Page 569

Getting a random sampling

5,5,2,9,1,0,4,5,3,1,2,4,1,9,4,6,6,9,1,7

5,5,2,9,1,0,4,5,3,1,2,4,1,9,4,6,6,9,1,7

55 29 10 45 31 24 19 46 69 17

5,5,2,9,1,0,4,5,3,1,2,4,1,9,4,6,6,9,1,7

55 29 10 45 31 24 19 46 69 17

Sample

65 64 68 65 63

63 64 62 64 67

Find the mean of the sample

65 64 68 65 63

63 64 62 64 67

Sample Mean

Mean = 62 + 2(63) + 3(64) + 2(65) + 67 + 68

10

Mean = 645

10

Mean = 64.5

Standard Deviation of the Sample

62 63 63 64 64 64 65 65 67 68

Standard Deviation of the Sample
• 63 63 64 64 64 65 65 67 68
Random Sample
• Mean = 64.5
• Standard deviation = 1.75
• Compare the sample to the mean and standard deviation of the entire population. (example 9.14)
• Compare our sample to the author’s sample. (example 9.14)
Normal Distribution
• The distribution of many populations form the shape of a “bell-shaped” curve and are said to be normally distributed.
• If a population is normally distributed, approximately 68% of the population lies within 1 standard deviation of the mean. About 95% within 2 standard deviations. About 99.7% within 3 standard deviations.
Normal Distribution Example
• Suppose the 200 grades of a certain professor are normally distributed. The mean score is 74. The standard deviation is 4.3.
• What whole number grade constitutes an A, B, C, D and F?
• Approximately how many students will make each grade?
A: 83 and above 200 students
• B: 79 – 82
• C: 70 – 78
• D: 66 – 69
• F: 65 and below
A: 83 and above 5 people
• B: 79 – 82 27 people
• C: 70 – 78 136 people
• D: 66 – 69 27 people
• F: 65 and below 5 people
Normal Distribution
• The graph of a normal distribution is symmetric about a vertical line drawn through the mean. So the mean is also the median.
• The highest point of the graph is the mean, so the mean is also the mode.
• The area under the entire curve is one.
Z Curve
• The scale on the horizontal axis now shows a z – Score.

Any normal distribution in standard form will have mean 0 and standard deviation1.

• 68% of the data will lie between -1 and 1.
• 95% of the data will lie between -2 and 2.
• 99.7% of the data will lie between -3 and 3.
Z- Scores
• By using a z-Score, it is possible to tell if an observation is only fair, quite good, or rather poor.
• EXAMPLE: A z-Score of 2 on a national test would be considered quite good, since it is 2 standard deviations above the mean.
• This information is more useful than the raw score on the test.
Z- Scores
• z – Score of a data is determined by subtracting the mean from the data and dividing the result by the standard deviation.
• z = x - µ

σ

62,62,63,64,64,64,64,66,66,66
• Mean = 64.1
• Standard deviation ≈ 1.45
• Convert these data to a set of z-scores.
62,62,63,64,64,64,64,66,66,66

62, 63, 64, 66

z-scores: -1.45, -0.76, -0.07, 1.31

Percentiles
• The percentile tells us the percent of the data that is less than or equal to that data.
Percentile in a sample:62,62,63,64,64,64,64,66,66,66
• The percentile corresponding to 63 is the percent of the data less than or equal to 63.
• 3 data out of 10 data = .3 = 30% of the data is less than or equal to 63.
• For this sample, 63 is in the 30th percentile.
Percentile in a Population
• Remember that the area under the normal curve is one.
• The area above any interval under the curve is less than one which can be written as a decimal.
• Any decimal can be written as a percent by multiplying by 100 (which moves the decimal to the right 2 places).
• That number would tell us the percent of the population in that particular region.
Percentiles
• Working through this process, we can find the percent of the data less than or equal to a particular data – the percentile.
• The z-score tells us where we are on the horizontal scale.
• Table 9.4 on pages 585 and 586 convert the z-score to a decimal representation of the area to the left of that data.
• By converting that number to a percent, we will have the percentile of that data.
If the z-score of a data in a normal distribution is -0.76,what is it’s percentile in the population?
• Table 9.4 page 585
• Row marked -0.7