A Closer Look at the NEW A Closer Look at the NEW High School Statistics High School Statistics

StandardsStandardsFocus on A.9 and AII.11Focus on A.9 and AII.11

K-12 Mathematics InstitutesK-12 Mathematics InstitutesFall 2010Fall 2010

A Closer Look at the NEW A Closer Look at the NEW High School Statistics High School Statistics

StandardsStandardsFocus on A.9 and AII.11Focus on A.9 and AII.11

K-12 Mathematics InstitutesK-12 Mathematics InstitutesFall 2010Fall 2010

Fall 2010

Vertical ArticulationVertical ArticulationVertical ArticulationVertical Articulation

5.16 The student will 5.16 The student will b) describe the mean as fair shareb) describe the mean as fair share

6.15 The student will6.15 The student willa) describe mean as balance pointa) describe mean as balance point

Algebra I SOL A.9 The student, given a set of Algebra I SOL A.9 The student, given a set of data, will interpret variation in real-world data, will interpret variation in real-world contexts and interpret mean absolute contexts and interpret mean absolute deviation, standard deviation, and z-deviation, standard deviation, and z-scores.scores.

5.16 The student will 5.16 The student will b) describe the mean as fair shareb) describe the mean as fair share

6.15 The student will6.15 The student willa) describe mean as balance pointa) describe mean as balance point

Algebra I SOL A.9 The student, given a set of Algebra I SOL A.9 The student, given a set of data, will interpret variation in real-world data, will interpret variation in real-world contexts and interpret mean absolute contexts and interpret mean absolute deviation, standard deviation, and z-deviation, standard deviation, and z-scores.scores.

2

Fall 2010

Vertical ArticulationVertical ArticulationVertical ArticulationVertical Articulation

AFDA.7 The student will analyze the AFDA.7 The student will analyze the normal distribution.normal distribution.

Algebra II SOL A.11 The student will Algebra II SOL A.11 The student will identify properties of the normal identify properties of the normal distribution and apply those distribution and apply those properties to determine probabilities properties to determine probabilities associated with areas under the associated with areas under the standard normal curve.standard normal curve.

AFDA.7 The student will analyze the AFDA.7 The student will analyze the normal distribution.normal distribution.

Algebra II SOL A.11 The student will Algebra II SOL A.11 The student will identify properties of the normal identify properties of the normal distribution and apply those distribution and apply those properties to determine probabilities properties to determine probabilities associated with areas under the associated with areas under the standard normal curve.standard normal curve.

3

Fall 2010

Before we start – just a little Before we start – just a little reminder about sigma notation reminder about sigma notation

and subscript notationand subscript notation

Before we start – just a little Before we start – just a little reminder about sigma notation reminder about sigma notation

and subscript notationand subscript notation

654321

6

1

xxxxxxxi

i

87654321i8

1i

4

Fall 2010

Mean of a Data Set Containing Mean of a Data Set Containing n Elements = n Elements = µµ

Mean of a Data Set Containing Mean of a Data Set Containing n Elements = n Elements = µµ

n

xxxxx

n

xn

n

ii

...43211

x = Sample mean

µ = Population mean5

Fall 2010

Mean ProblemMean ProblemMean ProblemMean Problem

Joe has the following test grades: Joe has the following test grades: 85, 80, 83, 91, 97 and 72. In order 85, 80, 83, 91, 97 and 72. In order to make the academic team he to make the academic team he needs to have an 85 average. With needs to have an 85 average. With one test yet to take, he wants to one test yet to take, he wants to know what score he will need on know what score he will need on that to have an 85 average.that to have an 85 average.

Joe has the following test grades: Joe has the following test grades: 85, 80, 83, 91, 97 and 72. In order 85, 80, 83, 91, 97 and 72. In order to make the academic team he to make the academic team he needs to have an 85 average. With needs to have an 85 average. With one test yet to take, he wants to one test yet to take, he wants to know what score he will need on know what score he will need on that to have an 85 average.that to have an 85 average.

6

Fall 2010

Solve for x:

857

729791838085

x

What score will “balance” the number line ?

72 8380 91 97

85

135 2 6

12

7

87

2

Fall 2010

A student counted the number of A student counted the number of players playing basketball in the players playing basketball in the Central Tendency Tournament Central Tendency Tournament

each day over its two week period.each day over its two week period.

A student counted the number of A student counted the number of players playing basketball in the players playing basketball in the Central Tendency Tournament Central Tendency Tournament

each day over its two week period.each day over its two week period.

Data Set#110, 30, 50, 60, 70, 30, 80, 10, 30, 50, 60, 70, 30, 80,

90, 20, 30, 40, 40, 60, 2090, 20, 30, 40, 40, 60, 20

8

Fall 2010

A student counted the number of A student counted the number of players playing basketball in the players playing basketball in the Dispersion Tournament each day Dispersion Tournament each day

over its two week period. over its two week period.

A student counted the number of A student counted the number of players playing basketball in the players playing basketball in the Dispersion Tournament each day Dispersion Tournament each day

over its two week period. over its two week period.

Data Set#250, 30, 40, 50, 40, 60, 50, 50, 30, 40, 50, 40, 60, 50,

40, 30, 50, 30, 50, 60, 5040, 30, 50, 30, 50, 60, 50

9

Fall 2010

How are the How are the two data two data

sets similar sets similar and how are and how are

they they different?different?

How are the How are the two data two data

sets similar sets similar and how are and how are

they they different?different?

Mean

Data Set #1

45

Data Set #2

4510

Fall 2010

How are the How are the two data two data

sets similar sets similar and how are and how are

they they different?different?

How are the How are the two data two data

sets similar sets similar and how are and how are

they they different?different?

XX Data Set #1 Data Set #2

1010 11 00

2020 22 00

3030 33 33

4040 22 33

5050 11 66

6060 22 22

7070 11 00

8080 11 00

9090 11 00

Frequency

Frequency (x)

11

Fall 2010

Data Set #1 0 10 20 30 40 50 60 70 80 90 100

x x x x x x x x x

x

x

x x

x

0 10 20 30 40 50 60 70 80 90 100

x

x x x

x

x

x

x

x

x

x x x

Data Set #2

Line Plot

12

Fall 2010

Mean = 45

10

30 30

50

60

70

8090

20

30

40 40

20

60

Data Set#1

Distance from the mean

14

Fall 2010

Mean = 45

10

30 30

50

60

70

8090

20

30

40 40

20

60

What if we find the average of the difference between each data value and the mean?

ix

n

-35

-15

5

1525

-15

3545

-25 -15

-5 -5

15

-25

15

Fall 2010

=0

ix

n

-35-15+5+15+25-15+35+45-25-15-5-5+15-25 14

What if we find the average of the difference between each data value and the mean?

16

Fall 2010

Mean = 45

10

30 30

50

60

70

8090

20

30

40 40

20

60

What if we find the average of the DISTANCES from each data value to the mean?

ix

n

35

15

5

1525

15

3545

25 15

5 5

15

25

17

Fall 2010

ix

n

35+15+5+15+25+15+35+45+25+15+5+5+15+25=

14

280 14

= 20

What if we find the average of the DISTANCES from each data value to the mean?

18

Fall 2010

Mean Absolute DeviationMean Absolute DeviationMean Absolute DeviationMean Absolute Deviation

1

n

ii

x

n

19

Fall 2010

Calculate the Calculate the Mean Mean

Absolute Absolute Deviation of Deviation of Data Set #2Data Set #2

Calculate the Calculate the Mean Mean

Absolute Absolute Deviation of Deviation of Data Set #2Data Set #2

X | X - μ |50 5

30 15

40 5

50 5

40 5

60 15

50 5

40 5

30 15

50 5

30 15

50 5

60 15

50 5Sum = 120

20

μ=45

Fall 2010

Mean Abs. Dev. = 57.814

120

21

Fall 2010

Mean = 45

10

30 30

50

60

70

8090

20

30

40 40

20

60

What if we find the average of the squares of the difference from each data value to the mean?

n

x 2i

35

15

5

1525

15

3545

25 15

5 5

15

25

22

Fall 2010

352+152+52+152+252+152+352+452+252+152+52+52+152+252=7550

7550 14

=539.286

Called the VARIANCE

n

x 2i What if we find the

average of the squares of the difference from each data value to the mean?

23

Fall 2010

Standard Deviation of a Standard Deviation of a Population Data SetPopulation Data Set

Standard Deviation of a Standard Deviation of a Population Data SetPopulation Data Set

2

1

n

ii

x

n

24

Fall 2010

Standard Deviation of Standard Deviation of Data Set #1Data Set #1

Standard Deviation of Standard Deviation of Data Set #1Data Set #1

539.286 23.222

25

Fall 2010

Mean = 45

10

30 30

50

60

70

8090

20

30

40 40

20

60

One Standard Deviation on either side of the Mean

26

Fall 2010

Population vs. Sample Standard Population vs. Sample Standard Deviation for Data Set #1Deviation for Data Set #1

Population vs. Sample Standard Population vs. Sample Standard Deviation for Data Set #1Deviation for Data Set #1

This is if the data set is the population.

Casio Texas Instruments

Population Standard DeviationSample Standard Deviation

27

Fall 2010

““Sample Standard Deviation” Sample Standard Deviation” and Bessel Adjustmentand Bessel Adjustment

““Sample Standard Deviation” Sample Standard Deviation” and Bessel Adjustmentand Bessel Adjustment

11

2

n

xxs

n

ii

28

Fall 2010

Standard Deviation Notation Standard Deviation Notation RecapRecap

Standard Deviation Notation Standard Deviation Notation RecapRecap

µ µ = mean of a population= mean of a population

σσ = population standard deviation = population standard deviation

s = sample standard deviation s = sample standard deviation (estimation of a population standard (estimation of a population standard deviation based upon a sample)deviation based upon a sample)

µ µ = mean of a population= mean of a population

σσ = population standard deviation = population standard deviation

s = sample standard deviation s = sample standard deviation (estimation of a population standard (estimation of a population standard deviation based upon a sample)deviation based upon a sample)

29

Fall 2010

How do the 2 data sets compare?How do the 2 data sets compare?How do the 2 data sets compare?How do the 2 data sets compare?

Data Set #1 Data Set #2

30

Fall 2010

Describing the position of data Describing the position of data relative to the mean.relative to the mean.

Describing the position of data Describing the position of data relative to the mean.relative to the mean.

- Can measure in terms of actual Can measure in terms of actual data distance units from the mean. data distance units from the mean.

- Measure in terms of standard Measure in terms of standard deviation units from the mean. deviation units from the mean.

- Can measure in terms of actual Can measure in terms of actual data distance units from the mean. data distance units from the mean.

- Measure in terms of standard Measure in terms of standard deviation units from the mean. deviation units from the mean.

ix

z-score standard measureix

31

Fall 2010

Why do that?Why do that?Why do that?Why do that?

So we can compare elements So we can compare elements from two different data sets from two different data sets relative to the position within relative to the position within their own data set.their own data set.

32

Fall 2010

Consider this problem…Consider this problem…Consider this problem…Consider this problem…

Amy scored a 31 on the Amy scored a 31 on the mathematics portion of her mathematics portion of her 2009 ACT2009 ACT®® ( (µ=21 µ=21 σσ=5.3).=5.3).

Stephanie scored a 720 on the Stephanie scored a 720 on the mathematics portion of her mathematics portion of her 2009 SAT2009 SAT®® ( (µ=515 µ=515 σσ=116.0).=116.0).

Amy scored a 31 on the Amy scored a 31 on the mathematics portion of her mathematics portion of her 2009 ACT2009 ACT®® ( (µ=21 µ=21 σσ=5.3).=5.3).

Stephanie scored a 720 on the Stephanie scored a 720 on the mathematics portion of her mathematics portion of her 2009 SAT2009 SAT®® ( (µ=515 µ=515 σσ=116.0).=116.0).

33

Fall 2010

Whose achievement was higher Whose achievement was higher on the mathematics portion of on the mathematics portion of their national achievement test?their national achievement test?

Whose achievement was higher Whose achievement was higher on the mathematics portion of on the mathematics portion of their national achievement test?their national achievement test?

Consider this problem…Consider this problem…Consider this problem…Consider this problem…

34

Fall 2010

Using z-scores to compareUsing z-scores to compareUsing z-scores to compareUsing z-scores to compare

AmyAmy

StephanieStephanie

AmyAmy

StephanieStephanie

35

1.89 vs. 1.77 What Does This Mean?

Fall 2010

By the end of Algebra I, we By the end of Algebra I, we have asked and answered the have asked and answered the

following BIG questions….following BIG questions….

By the end of Algebra I, we By the end of Algebra I, we have asked and answered the have asked and answered the

following BIG questions….following BIG questions….How do we quantify the central How do we quantify the central

tendency of a data set?tendency of a data set?How do we quantify the spread of a How do we quantify the spread of a

data set?data set?How do we quantify the relative How do we quantify the relative

position of a data value within a position of a data value within a data set?data set?

How do we quantify the central How do we quantify the central tendency of a data set?tendency of a data set?

How do we quantify the spread of a How do we quantify the spread of a data set?data set?

How do we quantify the relative How do we quantify the relative position of a data value within a position of a data value within a data set?data set?

36

Fall 2010

So what do Algebra I student need to be able to do?So what do Algebra I student need to be able to do?So what do Algebra I student need to be able to do?So what do Algebra I student need to be able to do?

A.9 DOE ESSENTIAL KNOWLEDGE AND SKILLSA.9 DOE ESSENTIAL KNOWLEDGE AND SKILLSThe student will use problem solving, mathematical communication, mathematical reasoning, The student will use problem solving, mathematical communication, mathematical reasoning,

connections, and representations toconnections, and representations to

- Analyze descriptive statistics to determine the implications Analyze descriptive statistics to determine the implications for the real-world situations from which the data derive.for the real-world situations from which the data derive.

- Given data, including data in a real-world context, calculate Given data, including data in a real-world context, calculate and interpret the mean absolute deviation of a data set. and interpret the mean absolute deviation of a data set.

- Given data, including data in a real-world context, calculate Given data, including data in a real-world context, calculate variance and standard deviation of a data set and interpret variance and standard deviation of a data set and interpret the standard deviation. the standard deviation.

- Given data, including data in a real-world context, calculate Given data, including data in a real-world context, calculate and interpret z-scores for a data set.and interpret z-scores for a data set.

- Explain ways in which standard deviation addresses Explain ways in which standard deviation addresses dispersion by examining the formula for standard dispersion by examining the formula for standard deviation.deviation.

- Compare and contrast mean absolute deviation and Compare and contrast mean absolute deviation and standard deviation in a real-world context.standard deviation in a real-world context.

A.9 DOE ESSENTIAL KNOWLEDGE AND SKILLSA.9 DOE ESSENTIAL KNOWLEDGE AND SKILLSThe student will use problem solving, mathematical communication, mathematical reasoning, The student will use problem solving, mathematical communication, mathematical reasoning,

connections, and representations toconnections, and representations to

- Analyze descriptive statistics to determine the implications Analyze descriptive statistics to determine the implications for the real-world situations from which the data derive.for the real-world situations from which the data derive.

- Given data, including data in a real-world context, calculate Given data, including data in a real-world context, calculate and interpret the mean absolute deviation of a data set. and interpret the mean absolute deviation of a data set.

- Given data, including data in a real-world context, calculate Given data, including data in a real-world context, calculate variance and standard deviation of a data set and interpret variance and standard deviation of a data set and interpret the standard deviation. the standard deviation.

- Given data, including data in a real-world context, calculate Given data, including data in a real-world context, calculate and interpret z-scores for a data set.and interpret z-scores for a data set.

- Explain ways in which standard deviation addresses Explain ways in which standard deviation addresses dispersion by examining the formula for standard dispersion by examining the formula for standard deviation.deviation.

- Compare and contrast mean absolute deviation and Compare and contrast mean absolute deviation and standard deviation in a real-world context.standard deviation in a real-world context.

37

Fall 2010

Let’s gather some dataLet’s gather some dataand calculate some statistics.and calculate some statistics.

Report your Report your heightheightto the nearest inch.to the nearest inch.

Let’s gather some dataLet’s gather some dataand calculate some statistics.and calculate some statistics.

Report your Report your heightheightto the nearest inch.to the nearest inch.

38

Fall 2010

Length of Length of Boys’ Boys’ Name Name

SummarySummary

Length of Length of Boys’ Boys’ Name Name

SummarySummary

#letters freq

1 0

2 1

3 10

4 71

5 137

6 153

7 89

8 26

9 9

10 2

11 2

12 0

13 0

14 0

total 500

http

://w

ww

.ssa

.gov

/OA

CT

/bab

ynam

es/

39

Fall 2010

StatisticsStatisticsStatisticsStatistics

Mean = Mean = 5.7465.746

Population Standard Population Standard Deviation = 1.3044Deviation = 1.3044

Sample Standard Sample Standard Deviation=1.3057Deviation=1.3057

Mean = Mean = 5.7465.746

Population Standard Population Standard Deviation = 1.3044Deviation = 1.3044

Sample Standard Sample Standard Deviation=1.3057Deviation=1.3057

40

Fall 2010

DistributionDistributionDistributionDistribution

0 1

10

71

137

153

89

26

92 2 0 0 0

0

20

40

60

80

100

120

140

160

180

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Nu

mb

er o

f bab

ies

Number of letters

Length of most popular Boy names in 2009

41

Fall 2010

Length of most popular Boy names in 2009

0

20

40

60

80

100

120

140

160

180

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Number of letters

Nu

mb

er

of b

ab

ies

500

71

500

137 500

153

500

89

500

26 500

9

500

2

500

2

500

1 500

10

What is the probability of selecting a name with exactly 6 letters?

What is the probability of selecting a name greater than or equal to 3 letters, but less than or equal to 9 letters?

What is the probability of selecting a name between 1 and 13 letters?

Make up a problem:

What is the probability of ________________?

Fall 2010

Let’s look at a distribution of Let’s look at a distribution of heights for a population.heights for a population.

Let’s look at a distribution of Let’s look at a distribution of heights for a population.heights for a population.

μ=68”

0.1995

71”

0.0648

prob

abili

ty

height

43

Fall 2010

Height as Continuous DataHeight as Continuous DataHeight as Continuous DataHeight as Continuous Data

0.1995

71”

0.0648

μ=68”

44

Fall 2010

Algebra II & Algebra II & Normal DistributionsNormal Distributions

Algebra II & Algebra II & Normal DistributionsNormal Distributions

45

Fall 2010

5 Characteristics of a 5 Characteristics of a Normal DistributionNormal Distribution

5 Characteristics of a 5 Characteristics of a Normal DistributionNormal Distribution

1. The mean, median and mode are equal.2. The graph of a normal distribution is

called a NORMAL CURVE.3. A normal curve is bell-shaped and

symmetrical about the mean.4. A normal curve never touches, but gets closer and closer to the x-axis as it gets farther from the mean.5. The total area under the curve is equal to

one.

46

Fall 2010

Examples of Normally Examples of Normally Distributed DataDistributed Data

Examples of Normally Examples of Normally Distributed DataDistributed Data

SAT scoresSAT scores

Height of 10-year-old boysHeight of 10-year-old boys

Weight of cereal in each Weight of cereal in each 24 ounce box24 ounce box

Tread life of tiresTread life of tires

Time it takes to tie your shoesTime it takes to tie your shoes

SAT scoresSAT scores

Height of 10-year-old boysHeight of 10-year-old boys

Weight of cereal in each Weight of cereal in each 24 ounce box24 ounce box

Tread life of tiresTread life of tires

Time it takes to tie your shoesTime it takes to tie your shoes

47

Fall 2010

The probability density function for normally distributed data can be written as a function of the mean, standard deviation,

and data values.

2

2

2

)(

22

1

x

ey

(x,y)=(data value, relative likelihood for that data value to occur)

48

Area under curve – up to a data value

( ) 0.5P x

Area under curve – from a data value to ∞

1( ) ???P x x 1x

1x 2x

1 2( ) ??P x x x

Area under curve – between two data values.

Fall 2010

68-95-99.7 Rule – Empirical Rule68-95-99.7 Rule – Empirical Rule68-95-99.7 Rule – Empirical Rule68-95-99.7 Rule – Empirical Rule

Do not underestimate the power of the quick sketch.52

Fall 2010

68-95-99.7 Rule – Empirical Rule68-95-99.7 Rule – Empirical Rule68-95-99.7 Rule – Empirical Rule68-95-99.7 Rule – Empirical RuleA normally distributed data set has A normally distributed data set has

µ=50 and µ=50 and σσ=5. What percent of the =5. What percent of the data falls between 45 and 55?data falls between 45 and 55?

A normally distributed data set has A normally distributed data set has µ=22 and µ=22 and σσ=1.5. What would be the =1.5. What would be the value of an element of this data set value of an element of this data set with z-score = 2? z-score = -2?with z-score = 2? z-score = -2?

A normally distributed data set has A normally distributed data set has µ=50 and µ=50 and σσ=5. What percent of the =5. What percent of the data falls between 45 and 55?data falls between 45 and 55?

A normally distributed data set has A normally distributed data set has µ=22 and µ=22 and σσ=1.5. What would be the =1.5. What would be the value of an element of this data set value of an element of this data set with z-score = 2? z-score = -2?with z-score = 2? z-score = -2?

53

Fall 2010

A machine fills 12 ounce Potato Chip bags. A machine fills 12 ounce Potato Chip bags. It places chips in the bags. Not all bags It places chips in the bags. Not all bags weigh exactly 12 ounces. The weight of the weigh exactly 12 ounces. The weight of the chips placed is normally distributed with a chips placed is normally distributed with a mean of 12.4 ounces and with a standard mean of 12.4 ounces and with a standard deviation 0.2 ounces. If you purchase a bag deviation 0.2 ounces. If you purchase a bag filled by this dispenser what is the filled by this dispenser what is the likelihood it has less than 12 ounces? likelihood it has less than 12 ounces?

A machine fills 12 ounce Potato Chip bags. A machine fills 12 ounce Potato Chip bags. It places chips in the bags. Not all bags It places chips in the bags. Not all bags weigh exactly 12 ounces. The weight of the weigh exactly 12 ounces. The weight of the chips placed is normally distributed with a chips placed is normally distributed with a mean of 12.4 ounces and with a standard mean of 12.4 ounces and with a standard deviation 0.2 ounces. If you purchase a bag deviation 0.2 ounces. If you purchase a bag filled by this dispenser what is the filled by this dispenser what is the likelihood it has less than 12 ounces? likelihood it has less than 12 ounces?

54

Fall 2010

Can you represent this as area Can you represent this as area under a normal curve?under a normal curve?

Can you represent this as area Can you represent this as area under a normal curve?under a normal curve?

12.412

Area=0.02275

55

Fall 2010

What fraction of the bags have between 12.1 and 12.5 ounces? Shade the region that represents that amount.

56

Fall 2010

Standard Normal DistributionStandard Normal DistributionStandard Normal DistributionStandard Normal Distribution

0

157

Fall 2010

Standard

Normal

Curve

0

58

Fall 2010

Normal Distributions can be Normal Distributions can be transformed into a Standard Normal transformed into a Standard Normal

Distribution using the z-score of Distribution using the z-score of corresponding data values.corresponding data values.

Normal Distributions can be Normal Distributions can be transformed into a Standard Normal transformed into a Standard Normal

Distribution using the z-score of Distribution using the z-score of corresponding data values.corresponding data values.

Example: 2010 SAT math scores for college bound seniors in VA

Mean=512 Standard Deviation=110

59College Board State Profile Report – Virginia (college bound seniors March 2010)

Fall 2010

MappingMappingMappingMappingSAT Score SAT Score Standard scoreStandard score

512 ( )512 ( ) 00

512+110 ( )512+110 ( ) 11

512 –110 ( )512 –110 ( ) -1-1

512+(2)110 ( )512+(2)110 ( ) 22

512 – (2)110 ( )512 – (2)110 ( ) -2-2

XXii

xxii – – μμ

σσ

2 2

z-score =

60

Fall 2010 61

z-scores below the mean

Fall 2010

Given the height of a population is normally distributed with a mean height = 68” with a standard deviation = 3.2”, what percent of the population is less than 61”?

z-score=

1875.22.3

6861

1875.2

2.3

6861

So, 1.43% of the population will be less than to 61”

Round to -2.19 for the z-table lookup.

62

Fall 2010

z-scores above the mean

63

Fall 2010

Using z-scores to compare Using z-scores to compare (revisited)(revisited)

Using z-scores to compare Using z-scores to compare (revisited)(revisited)

AmyAmy

StephanieStephanie

AmyAmy

StephanieStephanie

0.978697th percentile

0.961696th percentile

64

Fall 2010

So what do Algebra II students need to be able to do?So what do Algebra II students need to be able to do?So what do Algebra II students need to be able to do?So what do Algebra II students need to be able to do?

A2.11 DOE ESSENTIAL KNOWLEDGE AND SKILLSA2.11 DOE ESSENTIAL KNOWLEDGE AND SKILLSThe student will use problem solving, mathematical communication, mathematical reasoning, The student will use problem solving, mathematical communication, mathematical reasoning,

connections, and representations toconnections, and representations to

- Identify the properties of a normal probability Identify the properties of a normal probability distribution.distribution.

- Describe how the standard deviation and the mean Describe how the standard deviation and the mean affect the graph of the normal distribution.affect the graph of the normal distribution.

- Compare two sets of normally distributed data using a Compare two sets of normally distributed data using a standard normal distribution and z-scores.standard normal distribution and z-scores.

- Represent probability as area under the curve of a Represent probability as area under the curve of a standard normal probability distribution.standard normal probability distribution.

- Use the graphing calculator or a standard normal Use the graphing calculator or a standard normal probability table to determine probabilities or probability table to determine probabilities or percentiles based on z-scores.percentiles based on z-scores.

A2.11 DOE ESSENTIAL KNOWLEDGE AND SKILLSA2.11 DOE ESSENTIAL KNOWLEDGE AND SKILLSThe student will use problem solving, mathematical communication, mathematical reasoning, The student will use problem solving, mathematical communication, mathematical reasoning,

connections, and representations toconnections, and representations to

- Identify the properties of a normal probability Identify the properties of a normal probability distribution.distribution.

- Describe how the standard deviation and the mean Describe how the standard deviation and the mean affect the graph of the normal distribution.affect the graph of the normal distribution.

- Compare two sets of normally distributed data using a Compare two sets of normally distributed data using a standard normal distribution and z-scores.standard normal distribution and z-scores.

- Represent probability as area under the curve of a Represent probability as area under the curve of a standard normal probability distribution.standard normal probability distribution.

- Use the graphing calculator or a standard normal Use the graphing calculator or a standard normal probability table to determine probabilities or probability table to determine probabilities or percentiles based on z-scores.percentiles based on z-scores.

65

Fall 2010

ResourcesResourcesResourcesResources2009 Mathematics SOL and related resources 2009 Mathematics SOL and related resources

http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/review.shtml

Instructional docs including the technical Instructional docs including the technical assistance documents for A.9 and AII.11 assistance documents for A.9 and AII.11 http://www.doe.virginia.gov/instruction/high_school/mathematics/index.shtml

2009 Mathematics SOL and related resources 2009 Mathematics SOL and related resources http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/review.shtml

Instructional docs including the technical Instructional docs including the technical assistance documents for A.9 and AII.11 assistance documents for A.9 and AII.11 http://www.doe.virginia.gov/instruction/high_school/mathematics/index.shtml

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