Statistics Chapter 9

Day 1

Unusual Episode

MS133 Final Exam Scores

Line Plot or Dot Plot

Stem and Leaf

Ordered Stem and Leaf

Frequency Table GradeScoreTallyFrequency

Frequency Table GradeScoreTallyFrequency A90-100IIII5 B80-89IIII5 C70-79IIII IIII IIII14 D60-69IIII III8 F0-59III3

Bar Graph

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°

5 B’s out of 35 grades total ≈ 14% ≈ 51°

14 C’s out of 35 grades

14/35 =.4 = 40%.4 x 360° = 144°

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°

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°

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

Eisenhower High School Boys Heights

EHS Boys’ Heights HeightFrequencyRelative Frequency

EHS Boys’ Heights HeightFrequencyRelative Frequency

Eisenhower High School Boys Heights

EHS Boys’ Heights

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

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 Questions Page 538

Measures of Central Tendency Mean – “Evening-off” Median – “Middle” Most – “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 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 =

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 =

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 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 =

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 =

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 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 =

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 =

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: = 46 Class S:82 – 67 = 15 Class T:96 – 40 = 56

Class RClass SClass T Mean = Median = Mode =7769,73,7640 Range =461556

Weighted Mean Example 9.7 Owner/Manager earned $850,000 Assistant Manager earned $48, 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 SalaryFrequency $18,0003 $27,00016 $48,0001 $850,0001

MEAN Mean = 3(18,000)+16(27,000)+48, , = ≈ $65,905

MEDIAN SalaryFrequency $18,0003 $27,00016 $48,0001 $850,0001

MEDIAN SalaryFrequencyCumulative Frequency $18,00031 – 3 $27, $48, $850,000121

MODE SalaryFrequencyCumulative Frequency $18,00031 – 3 $27, $48, $850,000121

RANGE SalaryFrequencyCumulative Frequency $18,00031 – 3 $27, $48, $850,000121

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

Grade Point Average A weighted mean 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 First Semester Grades HoursGrade 3D 4F 2B 3C 2C 1A

Sally Ann’s First semester GPA to the nearest hundredth

Sally Ann’s Second Semester HoursGrade 3C 3C 3B 3B

Sally Ann’s Second Semester GPA

Sally Ann’s Cumulative GPA Total quality points earned Total hours attempted

Sally Ann’s New GPA to the nearest hundredth

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

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

Standard Deviation of Class X

Page 558 Example 9.11 Find the mean (to the nearest tenth): 35, 42, 61, 29, 39

Page 558 Example 9.11 Find the mean (to the nearest tenth): ≈ 41.2 Standard deviation (to the nearest tenth): 35, 42, 61, 29, 39

Page 558 Example 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 =78Lower 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 =78Lower Quartile = 71 Inner Quartile Range = 7IQR 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 =78Lower Quartile = 71 Inner Quartile Range = 7IQR 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 =78Lower Quartile = 71 Inner Quartile Range = 7IQR 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 =78Lower Quartile = 71 Inner Quartile Range = 7IQR 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 =78Lower Quartile = 71 Inner Quartile Range = 7IQR 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 = 76.5LQ = 70 IQR = 6.5IQR x 1.5 = 9.75 Checkpoints for outliers: 60.25, Outliers = noneWhisker 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 = 85LQ = 49 IQR = 36 IQR x 1.5 = 54 Checkpoints for Outliers: -5, 139 Outliers = noneWhisker Ends = 40, 96

Day 4

Homework Questions Page 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, Sample

Find the mean of the sample

Mean = (63) + 3(64) + 2(65)

Sample Mean Mean = (63) + 3(64) + 2(65) Mean = Mean = 64.5

Standard Deviation of the Sample

Standard Deviation of the Sample

Standard Deviation

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

68% of the data is within 1 standard deviation of the mean

95% of the data is within 2 standard deviations of the mean

99.7% of the data is within 3 standard deviations of the mean

Normal Distribution

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 above200 students B: 79 – 82 C: 70 – 78 D: 66 – 69 F: 65 and below

A: 83 and above 5 people B: 79 – people C: 70 – people D: 66 – 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

Standardized form of the normal distribution (z curve)

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 % 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 data out of 10 data =.3 = 30% of the data is less than or equal to 63. For this sample, 63 is in the 30 th 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 % 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 and 0.44

Interval Example Show that 34% of a normally distributed population lies between the z scores of and 0.44 Table 9.4, page % to the left of % to the left of % - 33% = 34%

Day 5

Homework Questions Page 576

Normal Distribution Lab

Day 6

Lab Questions

Statistics Review

M&M Lab