CCSS ALGEBRA 2 STATISTICS ESSENTIALS: The Stat Skills and Background You’ll Need Rob Gerver, Ph.D. North Shore HS Glen Head, NY [email protected]

CCSS ALGEBRA 2 STATISTICS ESSENTIALS:The Stat Skills and Background You’ll Need

Rob Gerver, Ph.D.North Shore HSGlen Head, NY

[email protected]

STAT TOPICS IN CCSS ALGEBRA 2:Content Standards S-ID, S-IC, S-CP, S-MD• Descriptive statistics: Interpreting categorical and

bivariate and univariate quantitative data• Probability: Conditional probability, independence,

expected value• Inferential Statistics: Sampling, making inferences,

and justifying conclusions

There is an emphasis on understanding, interpreting and critiquing—it’s not just students boxing numerical answers!

211 PAGES OF SAMPLE PROBLEMS!

https://www.engageny.org/resource/algebra-ii-module-4

https://webmail.northshoreschools.org/owa/redir.aspx?C=Q_qrMK91l02UXo9V-iEy-YUf1XZPy9II5w2TMUdvSCrcMNw2SXN4taKCsInFNuXJ2b1orBjGntQ.&URL=https://www.engageny.org/resource/algebra-ii-module-4

YOU’LL NEED TO KNOW MORE THAN JUST WHAT IS IN THE CURRICULUM!

CCSS ALGEBRA 2 STATISTICS ESSENTIALS

• Descriptive Statistics: Univariate Data• Descriptive Statistics: Bivariate Data• Probability and Independence• Inferential Statistics: Unbiased Estimators• Inferential Statistics: Sampling Distributions• Inferential Statistics: Experimental Design• Writing Projects• Balloon Help Tutorials

DISPLAYING UNIVARIATE DATA

Box and whisker plots, histograms, dot plots

UNIVARIATE STATISTICAL MEASURESCentral tendency:• Mean • Median-resistant to outliers• ModeDispersion: • Range—ignores spread except for high and low • Quartiles, IQR (colleges middle 50%)• Mean deviation, Mean absolute deviation• Variance, Standard deviation in descriptive and

inferential statistics: σ vs. s



BIVARIATE MEASURES• Correlation coefficient• Reg slope as a rate, interpreting r as strong,

moderate or weak, • Interpreting relationship as linear or not--

using scatterplot. • Making predictions--extrapolation

BIVARIATE NUANCES• Causation vs correlation• Lurking variables• Confounding variables

USE THE SCATTERPLOT! L1 L2 L3 L4 L5 L6

10 8.04 9.14 7.46 8 6.588 6.95 8.14 6.77 8 5.76

13 7.58 8.74 12.74 8 7.719 8.81 8.77 7.11 8 8.84

11 8.33 9.26 7.81 8 8.4714 9.96 8.1 8.84 8 7.04

6 7.24 6.13 6.08 8 5.254 4.26 3.1 5.39 19 12.5

12 10.84 9.13 8.15 8 5.567 4.82 7.26 6.42 8 7.915 5.68 4.74 5.73 8 6.89

Scatterplot x-value y-value Correlation Coefficient—3 decimal places

I L1 L2 .816II L1 L3 .816III L1 L4 .816IV L5 L6 .816

SCATTERPLOTS HELP TELL THE STORY!

I II III IV



PERMUTATIONS AND COMBINATIONS

• nCr and nPr

• Calculator commands• Using formulas by hand• Understanding “ordered” vs not• nCr gives number of samples without replacement

WHY IS PROBABILITY ALWAYS TEAMED WITH STATISTICS??!!

Probability is the basis of statistical inference.

THEORETICAL AND EMPIRICAL PROBABILITY

MAKE A CARD DECK POSTER!

Great for explaining independence and conditional probability!

UNDERSTANDING INDEPENDENCE

K = king; F= face; D = diamond

DECLARING INDEPENDENCE

The conditional probability definition is

If , A and B are independent events.

So a test for independence is

If A and B are disjoint events,

This formula holds if A and B are disjoint or not.

DISJOINT? EXHAUSTIVE? INDEPENDENT?

CONDITIONAL PROBABILITY USING TWO-WAY TABLES

Watches on Cable

Watches on Dish

Doesn’t Watch

TOTALS

Giant Fans 14 10 6 30Angel Fans 6 18 11 35Dodger Fans 4 20 5 29Athletics Fans 5 19 6 30Padre Fans 15 7 12 34TOTALS 44 74 40 158

Categorical (qualitative) variables cannot be ordered, but you can look at whether or not they are associated with each other; and if they are independent.

How California Baseball Fans Watch Their Team’s Games

CONDITIONAL PROBABILITY USING TWO-WAY TABLES

Smoking Status:

4-Year Degree

2-Year Degree

No Degree Totals

Current 51 22 43 116Former 92 21 28 141Never 68 9 22 99Totals 211 52 93 356

P(person is a former smoker) =P(person is a former smoker, given that they have no degree) =P(person has a degree) =P(person has a 4-year degree, given that they are a current smoker) =

Students need to have dexterity with the tables!

Students need to be able to use and interpret the algebraic formulas!

Conditional Probability Option: SIMPSON’S PARADOX

PLAYER PITCHER HITS AT-BATS AVERAGEJulie Righty 40 100 .400

Lefty 80 400 .200Jordan Righty 120 400 .300

Lefty 10 100 .100

Julie is better against righties and lefties, but Jordan is the better hitter overall (.260 vs 240)!! Unreal!

EXPECTED VALUEA carnival game called “Take Five” involves the rolling of a die. If it lands on 5, the winner gets $5. If it lands on 1, 2, or 3, the player receives $1. If it lands on 4 or 6, the player receives nothing. If the carnival organizers charge $2 to play this game, what is their expected profit if 1000 people play?

Die Face 1, 2, 3 5 4 or 6Payout X $1 $5 $0

Probability P(X) 1/2 1/6 1/3

E(x) = 1(1/2) + 5(1/6) + 0(1/3) = $1.33 and this is the average payout. (2 – 1.33)(1000) is expected profit.

EXPECTED VALUE

Age at Death 21 22 23 24 25

Profit X ?? ?? ?? ?? ?? ??

P(X) .00183 .00186 .00189 .00191 .00194 ????

A life Insurance company charges $250 annually for a $100,000 five-year term policy. What is their expected profit on this policy?

Age at Death 21 22 23 24 25

Profit X -99,750 -99,500 -99,250 -99,000 -98,750 +1,250

P(X) .00183 .00186 .00189 .00191 .00194 ????



THE MEAN: AN UNBIASED ESTIMATOR?

Is the average of all possible sample means the same as the actual population mean? If so, the mean would be an unbiased estimator. If not, the mean is biased. Let’s find out! Here is a population of 6 people’s scores: 1, 4, 5, 16, 17, 23. What is the population’s mean?_________

What is the mean of all the sample means?________Conjecture:

THE MEDIAN: AN UNBIASED ESTIMATOR?

Is the average of all possible sample medians the same as the actual population median? If so, the median would be an unbiased estimator. If not, the median is biased. Let’s find out! Here is a population of 6 people’s scores: 1, 4, 5, 16, 17, 23. What is the population’s mean?_________

What is the mean of all the sample medians?________Conjecture:

THE RANGE: AN UNBIASED ESTIMATOR?

Is the average of all possible sample ranges the same as the actual population range? If so, the range would be an unbiased estimator. If not, the range is biased. Let’s find out! Here is a population of 6 people’s scores: 1, 4, 5, 16, 17, 23. What is the population’s range?_________

What is the mean of all the sample ranges?________Conjecture:

THE SAMPLE VARIANCE s2: AN UNBIASED ESTIMATOR?Is the average of all possible sample variances the same as the actual population variance? If so, the sample variance would be an unbiased estimator. What is the population’s variance σ2?_________

What is the mean of all the sample variances?________Conjecture:

THE VARIANCE σ2: AN UNBIASED ESTIMATOR?Is the average of all possible variances the same as the actual population variance? If so, the sample variance would be an unbiased estimator. What is the population’s variance σ2?_________

What is the mean of all the σ2 variances from the samples?________Conjecture:

: AN UNBIASED ESTIMATOR OF p?The following Y’s and N’s are Yes/No responses to a question, from a population of 8:

Y, Y, Y, N, Y, N, Y, Y 1. What is the population proportion p, of Y’s?____

2. Imagine taking samples of size 2 with replacement. Make them here, using the grid. In each cell enter the proportion of Y’s in that sample.

3. What is the mean of all the sample proportions?_____4. Conjecture:

BIAS AND VARIABILITY:STRIVING FOR LOW BIAS AND LOW VARIALIBILITY

• Bias describes how near the sample statistics come to estimating the population parameter.

• Variability describes how scattered the sample statistics are.

• The sample mean and sample proportion have low bias and low variability.



What is a Density Curve?

• Area underneath curve, and above x-axis, is 1, representing 100%.

• Area under the curve, in any specified interval, represents a percent of the total area.

• Most famous density curve is the standard normal curve.

What is a Sampling Distribution?

• A density curve that represents a distribution of a selected statistic from all possible samples of a given size, taken from a specific population, with replacement.

• Area under any interval represents a percent of the samples.

• Most famous sampling distribution is the standard normal curve.

VIOLATING REPLACEMENTThe population of Oyster Bay is 293,214. Let’s say you wanted to select a sample of size 500 from this town. What is the probability, written as a fraction, that Bruno will be selected first? 1/293214 Convert this to an 8-place decimal carefully. .000003410478354 Let’s say 499 subjects were already picked and not replaced, and Bruno is not one of them. What is the probability he will be picked next, as a fraction? 1/292,715. Convert this to a decimal. .000003416292298. Compare the probably that Bruno was picked first to the probability he was picked 500th. What do you notice?_____. Since the population is so much larger than the sample size, Bruno’s probability is essentially the same whenever he is picked, giving us the independence we need to use the graphs and formulas. Large populations allow you to pick larger samples, and larger samples are usually more representative of the population.

CHOOSING SUBJECTS RANDOMLY

CHOOSING SUBJECTS RANDOMLY

If you “seed” your calculator,

you’ll get the same random

numbers.



EXPERIMENTAL DESIGN BASICS

• Control-set up a comparison group.• Replication—use high n for samples and also

repeat experiment in different settings.• Randomness—use correct sampling technique,

reliable and valid instruments, no lurking or confounding variables, correct design.

• Factor-the treatment• Response variable—quantified result after trt• Independence—subjects picked independently

THE LANGUAGE OF EXPERIMENTAL DESIGNDescriptive statistics Inferential statistics Population SampleObservational study LimitationsExperimental study ControlDelimitations SimulationSampling CensusVoluntary response sample Convenience sampleNonresponse Learning effect Undercoverage Placebo Matched pairs Double-blindHawthorne effect Randomness Replication Placebo effect Statistical significance

THE LOGIC BEHIND HYPOTHESIS TESTING

Use binomial theorem one-die roll example:

Binompdf(50, 1/6, 21) = .00001551

What do you choose to believe?Are you ever 100% sure you are correct?

HYPOTHESIS TESTS• Null hypothesis—the hypothesis of “no

difference.”• A sampling distribution is created, based on

the null hypothesis.• A sample is taken.• Data from the sample is analyzed as probable,

or improbable when compared to the sampling distribution.

CONFIDENCE INTERVALS• To explore, get a handle on, some unknown

numerical quantity.• Interval estimates vs. point estimates.• Build a margin of error around a sample statistic.• Margin of error based on sample size.• Increase n or lower confidence to shrink interval.• Can follow up a hypothesis test when null

hypothesis is rejected.

LIMITATIONS AND DELIMITATIONS• Limitations—time, money, effort, accessibility,

geographical proximity, release time from work. Limitations are imposed on you.

• Delimitations—deliberate limitations you impose on your experiment.

TYPES OF SAMPLES• SRS-Simple random sample-each sample has the

same chance of being selected.• Systematic Random Sample----stadium view,

school room heat• Stratified random sample-mimic population %’s• Cluster sample• Convenience (opportunistic sample)

COMPLETELY RANDOMIZED DESIGNThe completely randomized design takes a randomly-selected group of subjects and splits them randomly into groups that received different treatments.

CRITIQUING STUDIES• Matched pairs vs. two sample designs: Which is

preferable? When is matched pairs impossible?• Learning effect• Hawthorne effect• Poor sampling--design, sample size, instruments• Poor design• Violating assumptions of statistical test• Lurking variables• Confounding variables• Influential points• Outliers and resistance





What Are Balloon-Help Tutorials?• Designed to gradually break that old

math habit—”boxing” numerical answers devoid of any verbal explanation.• They require students to explain

selected (or all) aspects of a solution to a problem; enhancing it with anything they feel helps explain the problem.

Benefits of Balloon-Help Tutorials• Gets students in the habit of writing original, complete sentences

more often.• “If you can’t say it, you don’t know it.”• Gets the writing practice frequent, consistent, and spaced out

through the year.• Writing practice translates to better free response answers.• An alternative form of assessment.• The grade from these projects can be used in many ways.• Can be used for extra credit options, pinpointed on specific

student trouble areas.• Makes for a great showcase or bulletin board. • By-product of trying to teach them the writing skills is they learn

they math they are working on.

Excerpt from Sample Annotation Written By Students

“High correlation does not imply any causation. In the example with the number of drownings correlated to ice-cream sales, we found that each of those variables was highly correlated with the temperature. The relationship between ice cream sales and temperature is probably causal, as is the relationship between # drownings and temperature.”

Assessing the Projects

Although they are graded for mathematical accuracy, and creativity of annotations, students are welcome to employ their artistic side. However, color is to be used to improve the explanation of a mathematical point; not for the sake of “glitz.”

BALLOON HELP TUTORIALS GRADING SHEET: 1 – 10 in each category1._____The mathematics is correct.2._____The full-sentence explanations are correct.3._____The topic/problem is comprehensive and complete. 4._____All crucial points are addressed verbally.5._____Color is used with discretion to improve the explanation of a statistical point.6._____Mathematical and statistical notation and terminology are used correctly.7._____Captions for figures are descriptive and formatted correctly.8._____Table headings are descriptive and formatted correctly.9._____The physical layout of the project--text, diagrams, tables—are high quality.10._____Appropriate and sufficient examples are given.11._____Diagrams and/or tables are graduated where necessary.12._____The project does a clearer job of explaining the topic than the original notes do.13._____The depth and quality of the project are commensurate with the student’s ability.

Documents

CCSS ALGEBRA 2 STATISTICS ESSENTIALS: The Stat Skills and Background You’ll Need Rob Gerver, Ph.D. North Shore HS Glen Head, NY [email protected]