data (not just a lot of numbers)

DATA(Not Just a Lot of Numbers)

James Stewart Lothar Redlin Saleem Watson

College AlgebraA Course in Crisis?

Introductory collegiate mathematics is in the midst of a revolution…

-Nancy Baxter Hastings,

Dickinson College

Traditional College Algebra is a boring, archaic, torturous course that does not help students solve problems or become better citizens. It turns off students and discourages them from seeking more mathematics learning.

- Chris Arney,

Dean of Science and Mathematics, St. Rose College

College AlgebraA Course in Crisis?

NSF conference on “Rethinking the Courses Below Calculus” in Washington D.C in 2001. Some of the major themes to emerge from this conference:

• Spend less time on algebraic manipulation and more time on exploring concepts

• Reduce the number of topics but study those topics covered in greater depth

• Give greater priority to data analysis as a foundation for mathematical modeling

• Emphasize the verbal, numerical, graphical and symbolic representations of mathematical concepts

WHY DATA?

Over the past two decades computers have transformed public discourse by generating piles of data and myriad analyses of these data. Ordinary citizens must deal with numbers and data every day.

-Bernard Madison, University of Arkansas

Virtually any educated individual will need the ability: 1. to examine a set of data and recognize a behavioral pattern in it,2. to assess how well a given functional model matches the data,3. to recognize the limitations in the model,4. to use the model to draw appropriate conclusions,5. to answer approriate questions about the phenomenon being studied.

-Sheldon Gordon, Farmingdale State University of New York

WHY DATA?

• Data relate the real world and algebraic equations.

• Students can collect data and make models from data.

• Students can see how the model gives us information about the thing being modeled.

WHY DATA?

Greater Depth:

Connecting

the

Concepts

COLLECTING DATA

• From classmates (measurements)

Age, height, hand span, shoe size, hat size

COLLECTING DATA• From classmates (Surveys)

Survey

1. What is the value (in cents) of the coins in your pocket or purse? __________

2. How far is your daily commute to school (in miles)? _________

3. How many siblings do you have (including yourself)? __________

4. How many hours a week do you spend on the Internet? __________

5. How many hours a week do you spend on homework? ________

6. Rate your happiness. not happy happy very happy

7. Rate your satisfaction with your school work. not satisfied satisfied very satisfied

COLLECTING DATA

• From simple experiments– Bridge science

COLLECTING DATA

• From simple experiments– How quickly can you name your favorite things

– How many words can you recall from a memorized list (after a day, a week, a month).

Listing vegetables Memorizing a list

COLLECTING DATA

• From simple experiments– How quickly does water leak from a tank? Toricelli’s

Law

Toricelli’s Law The experiment Students performing the experiment

COLLECTING DATA

• From simple experiments– Radioactive decay—modeled with pennies

Radioactive DecayCoin Experiment

COLLECTING DATA

• From the Internet– How many farms in the US?

Farming in the 19th century Farming in the 20th century

COLLECTING DATA

• From the Internet– Population

Las Vegas 1900 Las Vegas 2000

• From Journal Articles– Algebra and Alcohol

Time (hr)

15 ml 30 ml 45 ml 60 ml

0. 0. 0. 0. 0.

0.067 0.032 0.071 — —

0.133 0.096 0.019 — —

0.167 — — 0.28 0.30

0.2 0.13 0.25 — —

0.267 0.17 0.30 — —

0.333 0.16 0.31 0.42 0.46

0.417 0.17 — — —

0.5 0.16 0.41 0.51 0.59

0.667 — — 0.61 0.66

0.667 — — 0.61 0.66

0.667 — — 0.61 0.66

0.667 — — 0.61 0.66

Concentration (mg/ml) after

95% ethanol oral dose

A

( ) tS t at b

COLLECTING DATA

DATA

15

15

64

11

15

22

30

25

15

15

83

77

346

15

76

32

711

131

22

4564

15

15

15

154515

74

8

175

0 321 812 573 224 355 22

10 546

11 457

12 673

13 752

14 375

15 972

10 436 5611 147 7612 583 34

10 478 56 74

11 547 76 80

12 103 34 24

GOAL: Get Information from Data

Model

Equation Scatter Plot

Regression

Matrix

Sample

Test of Hypothsis

Confidence Interval

Frequency HistogramMean

Standard Deviation

Proportion

Getting Information from Data

• Descriptive InformationTells us something about the data itself

• Inferential InformationTells us how to extend the information obtained from the data beyond the domain of the data.

The FORM of the Data

How does the data obtained from this survey differ for different questions?

• What is your age?• What is your height?• What is your hair color?• From which source do you mostly obtain the news?• Do you believe that the Universe began in a huge

explosion?

The Form of the Data

The form of the data tells us the kind of information we can obtain.

• One-Variable Data

• Two-Variable Data

• Categorical Data

• Sample Data

One-Variable Data

Age (yr) 2 2 2 3 3 3 4 4 4 4 4 5

Income ( thousands of dollars) 280 56 59 62 51

Selling Price (X 1000) 159 193 167 172 169 216 169 172

Descriptive Information

• Summary statistics:Central tendency: Mean, median

Dispersion: variance, standard deviation

One-Variable Data

Example: height of students Mean: 60”, S.D.: 10”

Given this information, which picture is more likely?

One-Variable Data


• Frequency histogram

Graphical, gives more complete information—tells how the data is distributed.

One-Variable Data


• Frequency histogram

One-Variable Data

Two-Variable Data

Age (yr) 2 2 2 3 3 3 4 4 4 4 4 5

Height (in) 32 31 36 38 35 41 47 43 42 38 39 45

Hourssince 6:00 am

Temperature(°F)

0 59

2 62

4 68

6 65

8 58

10 60

12 62

Depth (ft)

Pressure (lb/in²)

0 14.7

10 19.2

20 23.7

30 28.2

40 32.7

50 37.2

60 41.7


• Scatter plotGives description of the relationship between the variables.

• Regression Line (or other curve)Gives the curve that best fits (or best describes) the data

Two-Variable Data

Descriptive information

• Depth/Pressure Data

Two-Variable Data

Depth(ft)

Pressure(lb/ft2)

0 14.7

10 19.2

20 23.7

30 28.2

40 32.7

50 37.2


• TV Hours/BMI

Two-Variable Data

Hours TV BMI

0 15

0 17

.5 15

.5 18

.75 16

1 16

1 15

1 17

1.25 18

1.5 19

: :

Two-variable data(Goal: Find a relationship between the variables)


• Regression Line (or other curve)Gives the curve that best fits (or best describes) the data

Two-Variable Data

Depth / Pressure Hours TV / BMI


Inferential Information

• Regression Line (or other curve)Use the curve to get information not in the data (extrapolation or interpolation using the regression curve).

Two-Variable Data


Inferential Information

• Regression Line (or other curve)

Two-Variable Data

interpolateextrapolate

interpolate

extrapolate

a


• Tire Inflation-Tire Life Relation– Quadratic functions

Two-Variable Data

Tire Pressure/Tire Life

Pressure(lb/in2)

Tire life(mi X1000)

26 50

28 66

31 78

35 81

38 74

42 70

45 5920

20.24324 17.627 239.47y x x

QuadReg y=ax2+bx+c a=-.24324 b= 17.627 c= 239.47

a


• Species-Area Relation– Power functions

Two-Variable Data

Species-area data

CaveArea(m2)

Numberof species

La Escondida 18 1

El Escorpion 19 1

El Tigre 58 1

Mision Imposible 60 2

San Martin 128 5

El Arenal 187 4

La Ciudad 344 6

Virgen 511 7

550

550

PwrRegy = a * x ^ b

a = 0.140019

b = 0.640512

0.640.14y x

a


• Length-at-Age Relation– Polynomial functions

Two-Variable Data

(a)

Length-at-age data

Age(years)

Length(inches)

Age

(years)

Length(inches)

1 4.8 9 18.2

2 8.8 9 17.1

2 8.0 10 18.8

3 7.9 10 19.5

4 11.9 11 18.9

5 14.4 12 21.7

6 14.1 12 21.9

6 15.8 13 23.8

7 15.6 14 26.9

8 17.8 14 25.1

90 year old rock fish3 20.0155 0.372 3.95 1.21y x x x


• Algebra and alcohol– Surge Functions

Two-Variable Data

Time (hr)

15 ml 30 ml 45 ml 60 ml

0. 0. 0. 0. 0.

0.067 0.032 0.071 — —

0.133 0.096 0.019 — —

0.167 — — 0.28 0.30

0.2 0.13 0.25 — —

0.267 0.17 0.30 — —

0.333 0.16 0.31 0.42 0.46

0.417 0.17 — — —

0.5 0.16 0.41 0.51 0.59

0.667 — — 0.61 0.66

0.667 — — 0.61 0.66

0.667 — — 0.61 0.66

0.667 — — 0.61 0.66

Concentration (mg/ml) after

95% ethanol oral dose

A

( ) tS t at b

Categorical Data

Student Number Hair color Eye Color

1 Dark Brown

2 Blond Brown

3 Dark Blue

Results of survey:

These data need organizing!

Categorical data(Goal: Organize the data/Get information)


• Organize Data in a Matrix

Categorical Data

Hair ColorBlondeDark

Blue

Green

Brown

Red

9 3 0

0 6 3

3 1 1



• Organize Data in a Proportionality Matrix

Categorical Data


Blue

Green

Brown

Red

.75 .30 .00

.00 .60 .75

.25 .10 .25


Get information from the data

• Using matrix multiplication

Categorical Data

.75 .30 .00 500 615

.00 .60 .75 800 930

.25 .10 .25 600 355


Blue

Green

Brown

Red


Get information from the data

• Using matrix multiplication

Categorical Data

500 X .75 + 800 X .30 + 600X .00 = 615

Dark hair

Proportionblue eyes

Blond hair

Proportionblue eyes

Red hair

Proportionblue eyes

Sample Data

We sample the wine. (We don’t drink the whole bottle and then decide that the wine is no good.)

Get information about a population from a sample.

GOAL: Get Information from Data

• No information if the sample is not randomExample: Samples of height

Take sample from the basketball team

Example: Proportion of male to female students

Take sample from the girls dormitory

• No information if the sample size is too smallExample: A sample of one

Sample Data

Examples• Hypothesis: Equal number of male and female

students

A random sample of 30 students are all femaleConclusion: Reject hypothesis

• Hypothesis: A coin is fair

The coin is tossed 20 times and results in 20 headsConclusion: Reject hypothesis

Intuitive basis for inference from a sample

Alternate examples• Hypothesis: Equal number of male and female

students

A random sample of 30 students, 21 are femaleConclusion: Reject hypothesis?

• Hypothesis: A coin is fair

The coin is tossed 20 times, 16 headsConclusion: Reject hypothesis?

Intuitive basis for inference from a sample

• Tossing 30 coins is a binomial distribution

Statistical basis for inference from a sample

Number of Heads

The Mean orexpecten numberof heads

Our result

• The average height of male students in samples of 500 students is a normal distribution.

The Mean orexpected averageheight

Our result

How do we make these intuitive decisions? Because we know that some events are less likely than others. We intuitively “know” the probability distribution of certain events.

• For more refined estimates we need to know the probability distribution more accurately.


Decision Rule: If the probability of getting the sample we actually got (or a more extreme sample) is very small (say .05 or less), we reject our hypothesis.

We use the calculator for graphing, for regression, for matrix operations, etc…

So let’s use the calculator to find probabilities.


Hypothesis: Proportion of females in population is 0.6.


1-Proportion Z Test

p0 : .6

Sucesses,x: 70

n : 100

Alternate Hyp : Prop = p 0/

Random sample of 100 has 70 females

Random sample of 50has 35 females

1-Proportion Z Test

p0 : = .6

Z: = 2.04124

P-value : = .04124

P-hat : = .7

n : = .7

P-Value .04 <.05Reject Hypothesis

1-Proportion Z Test

p0 : .6

Sucesses,x: 35

n : 50

Alternate Hyp : Prop = p 0/

P-Value .14 >.05Fail to reject Hypothesis

1-Proportion Z Test

p0 : = .6

Z: = 1.4438

P-value : = .148915

P-hat : = .7

n : = .7

Hypothesis: Mean height of male students is 70”.


Random sample of 20 has mean height 70.13

Random sample of 6has mean height 72.6

P-Value .69 >.05Fail to reject Hypothesis

P-Value .01 <.05Reject Hypothesis

T Test

70

List: list1

Freq : 1

Alternate Hyp : /

T Test

0 = 70t = .400P-value = .694

df = 19

x = 70.13

/

_

Sx = 1.45n = 20

T Test

70

List: list2

Freq : 1

Alternate Hyp : /

T Test

0 = 70t = 4.026P-value = .010

df = 5

x = 72.63

/

_

Sx = 1.60n = 6

Research articles report results in terms of p-values


data (not just a lot of numbers)

Documents

data analysis

piles of data

set of data

torturous course

mathematics learning

given functional model

mathematical modelingemphasize

nsf conference