Statistics Chapter 9. Day 1 Unusual Episode MS133 Final Exam Scores 7986796578 9178948875 7153959679...
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Transcript of Statistics Chapter 9. Day 1 Unusual Episode MS133 Final Exam Scores 7986796578 9178948875 7153959679...
StatisticsChapter 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
9 1 1 4 5 6
8 6 8 9 6 4
7 9 1 8 8 9 7 9 8 5 9 7 4 8 7
6 2 9 3 5 7 4 9 8
5 3 8
4 9
Ordered Stem and Leaf
9 1 1 4 5 6
8 4 6 6 8 9
7 1 4 5 7 7 7 8 8 8 8 9 9 9 9
6 2 3 4 5 7 8 9 9
5 3 8
4 9
Frequency Table
Grade Score Tally Frequency
Frequency Table
Grade Score Tally Frequency
A 90-100 IIII 5
B 80-89 IIII 5
C 70-79 IIII IIII IIII 14
D 60-69 IIII III 8
F 0-59 III 3
Bar Graph
FREQUENCY
GRADES
MS133 Final Exam Grades
F'sD'sC'sB'sA's
14
12
10
8
6
4
2
Make a Pie Chart
• 5 A’s out of how many grades total?
• 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°
A's14%
• 5 B’s out of 35 grades total ≈ 14% ≈ 51°
A's14%
B's14%
A's14%
• 14 C’s out of 35 grades
• 14 C’s out of 35 grades
• 14/35 = .4 = 40%
• .4 x 360° = 144°
B's14%
A's14%
C's40%
B's14%
A's14%
• 8 D’s out of 35 grades total
• 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°
C's40%
B's14%
A's14%
D's23%
C's40%
B's14%
A's14%
• 3 F’s out of 35 total
• 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°
D's23%
C's40%
B's14%
A's14%
MS133 Final Exam Grades
F's 9%
D's23%
C's40%
B's14%
A's14%
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
Profit 8%
5% Maintenance3%
Equipment
Materials 37%
Labor47%
Histogram
• Table 9.2 Page 527
Eisenhower High School Boys Heights
7371696765 747270686664
HEIGHTS (inches)
18
14
10
6
2
FREQUENCY
EHS Boys’ Heights
Height Frequency Relative
Frequency
64 1
65 1 70 14
66 3 71 10
67 7 72 6
68 15 73 2
69 19 74 2
EHS Boys’ Heights
Height Frequency Relative
Frequency
64 1 .0125
65 1 .0125 70 14 .175
66 3 .0375 71 10 .125
67 7 .0875 72 6 .075
68 15 .1875 73 2 .025
69 19 .2375 74 2 .025
Eisenhower High School Boys Heights
.25
.20
.15
.10
.05
FREQUENCY
7371696765 747270686664
HEIGHTS (inches)
RELATI VE
EHS Boys’ Heights
HEIGHTS (inches)
FREQUENCY
7473727170696867666564
18
14
10
6
2
Day 2
Measures of Central Tendency Lab
Print your first name below.
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 (extra) x – y 2
• x – y (extra) x – y 2
• y + x – y
2
• 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
Homework QuestionsPage 538
Measures of Central Tendency
• Mean – “Evening-off”
• Median – “Middle”
• Most – “Most”
Class R
71 71 7679 77 76 7072 92 74 8679 46 79 7281 67 77 7277 63 77 6176
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Mean = Sum of all grades
Number of grades
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Mean = Sum of all grades
Number of grades
Mean = 1771
24
46,61,63,67,70,71,71,72,72,72,74,76,76,76,77,77,77,77,79,79,79,81,86,92
Mean = Sum of all grades
Number of grades
Mean = 1771
24
8.7324
1771x
Class S
72 77 75 7567 76 69 7671 68 77 7982 73 69 7668 69 71 7872 79 74 7373
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
Mean =
67,68,68,69,69,69,71,71,72,72,73,73,73,74,75,75,76,76,76,77,77,78,79,79,82
Mean = 25
1839
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
Mean = 25
1839
6.7325
1839x
Class T
74 79 86 8440 82 40 6140 49 70 8549 40 45 9174 96 81 8586 75 89 85
40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96
Mean =
40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96
Mean = 24
1686
40,40,40,40,45,49,49,61,70,74,74,75,79,81,82,84,85,85,85,86,86,89,91,96
Mean = 24
1686
3.7024
1686x
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 dispersion Greatest - 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 R Class S Class T
Mean = 73.8 73.6 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
Salary
$18,000
$27,000
$48,000
$850,000
MEAN
Salary Frequency
$18,000 3
$27,000 16
$48,000 1
$850,000 1
MEAN
Mean = 3(18,000)+16(27,000)+48,000+850,000
21
= 1384000 21
≈ $65,905
MEDIAN
Salary Frequency
$18,000 3
$27,000 16
$48,000 1
$850,000 1
MEDIAN
Salary Frequency Cumulative Frequency
$18,000 3 1 – 3
$27,000 16 4 - 19
$48,000 1 20
$850,000 1 21
MODE
Salary Frequency Cumulative Frequency
$18,000 3 1 – 3
$27,000 16 4 - 19
$48,000 1 20
$850,000 1 21
RANGE
Salary Frequency Cumulative Frequency
$18,000 3 1 – 3
$27,000 16 4 - 19
$48,000 1 20
$850,000 1 21
• Mean = $65,905
• Median = $27,000
• Mode = $27,000
• Range = $832,000
Grade Point AverageA weighted mean
quality points earned
hours attempted
Quality PointsEvery 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 First Semester Grades
Hours Grade
3 D
4 F
2 B
3 C
2 C
1 A
Sally Ann’s First semester GPAto the nearest hundredth
53.115
23
Sally Ann’s Second Semester
Hours Grade
3 C
3 C
3 B
3 B
Sally Ann’s Second Semester GPA
5.212
30
Sally Ann’s Cumulative GPA
Total quality points earned
Total hours attempted
Sally Ann’s New GPAto the nearest hundredth
96.127
53
Day 3
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
15
)90(2)85(2)82(4)78(2)72(3)60(2
Mean
60, 60, 72, 72, 72, 78, 78, 82, 82, 82, 82, 85, 85, 90, 90
15
)90(2)85(2)82(4)78(2)72(3)60(2
7815
1170
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
Standard Deviation of Class X
15
)9078(2)8578(2)8278(4)7878(2)7278(3)6078(2 222222
15
)12(2)7(2)4(4)0(2)6(3)18(2 222222
15
)12(2)7(2)4(4)0(2)6(3)18(2 222222
15
)144(2)49(2)16(4)0(2)36(3)324(2
15
)12(2)7(2)4(4)0(2)6(3)18(2 222222
15
)144(2)49(2)16(4)0(2)36(3)324(2
15
28898640108648
15
)12(2)7(2)4(4)0(2)6(3)18(2 222222
15
)144(2)49(2)16(4)0(2)36(3)324(2
15
28898640108648
97.84.8015
1206
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 =
100908070605040
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 =
**
100908070605040
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
**
100908070605040
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
**
100908070605040
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
**
100908070605040
Day 4
Homework QuestionsPage 561
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 3124 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 3124 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
Mean = 62 + 2(63) + 3(64) + 2(65) + 67 + 68
10
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
62 63 63 64 64 64 65 65 67 68
10
)685.64()675.64()655.64(2)645.64(3)635.64(2)625.64( 222222
Standard Deviation
10
)685.64()675.64()655.64(2)645.64(3)635.64(2)625.64( 222222
10
)5.3()5.2()5.(2)5(.3)5.1(2)5.2( 222222
Standard Deviation
10
)685.64()675.64()655.64(2)645.64(3)635.64(2)625.64( 222222
10
)5.3()5.2()5.(2)5(.3)5.1(2)5.2( 222222
10
25.1225.6)25(.2)25(.3)25.2(225.6
Standard Deviation
10
)685.64()675.64()655.64(2)645.64(3)635.64(2)625.64( 222222
10
)5.3()5.2()5.(2)5(.3)5.1(2)5.2( 222222
10
25.1225.6)25(.2)25(.3)25.2(225.6
75.105.310
5.30
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)
Beans or Fish
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 Curve
x + 3sx - 3s x - 2s x + 2sx - s x + sx
68% of the data is within 1 standard deviation of the mean
x - s x + sx
< >68%
95% of the data is within 2 standard deviations of the mean
x - 2s x + 2sx
< >95%
99.7% of the data is within 3 standard deviations of the mean
x - 3s x + 3sx
< >99.7%
Normal Distribution
x + 3sx - 3s
99.7%
95%
68%
x - 2s x + 2sx - s x + sx
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?
74x 3.4.. ds students200
61.1 65.4 69.7 86.982.678.374
• A: 83 and above 200 students• B: 79 – 82• C: 70 – 78• D: 66 – 69• F: 65 and below
61.1 65.4 69.7 86.982.678.374
• 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.
Normal Distribution
x + 3sx - 3s x - 2s x + 2sx - s x + sx
Standardized form of the normal distribution (z curve)
-3 -2 -1 3210
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• Column headed .06• Entry .2236• 22.36% of the population lies to the left of -0.76
Note the difference in finding the percentile in a sample and the
entire population.
Interval Example
• Show that 34% of a normally distributed population lies between the z scores of -0.44 and 0.44
Interval Example
• Show that 34% of a normally distributed population lies between the z scores of -0.44 and 0.44
• Table 9.4, page 585
• 33% to the left of -0.44
• 67% to the left of 0.44
• 67% - 33% = 34%
Day 5
Homework QuestionsPage 576
Normal Distribution Lab
Day 6
Lab Questions
Statistics Review
M&M Lab