Mr. Rogers - AP Statistics Objectives
 Syllabus 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter Probability Binomal Distr Sampling Distr Conf Intervals
 Unit Plan Practice Test Practice Test Answers
Latin
 Latin/Greek Root Words arch--------->ancient, example: archtype;         chrono------>time, example: chronology;             -dom----------->quantity/state, example: freedom               fer-------->carry, example: transfer;               gen--------->birth, example: generate;                 luc-------->light, example lucid;                 neo--------->new, example: neonatologist;                olig--------->few, example: oligarchy;              omni--------->all, omniscient;            sym--------->together, symbol; (Statistics connection)

Chap. 6 Binomial and Geometric Distributions

AP Statistics Standards

III. Anticipating Patterns: (continued)

A. Probability

1. Interpreting probability, including long-run relative frequency interpretation

2. “Law of Large Numbers” concept

3. Discrete random variables and their probability distributions, including binomial and geometric

4. Simulation of random behavior and probability distributions

5. Mean (expected value) and standard deviation of a random variable, and linear transformation of a random variable

B. Combining independent random variables

1. Notion of independence versus dependence

2. Mean and standard deviation for sums and differences of independent random variables

Objectives

 Essential Question: Can humans simulate a random process and why is this an important issue?

Random Variables

1. Be as one with the following vocabulary:

discrete distribution: A density curve (theoretical model of a probability distribution) that has a finite number of possible values within any finite segment of its range of values. Discrete distributions tend to look like they are made of stair steps. Examples include the binomial and geometric distributions.

continuous distribution: A density curve (theoretical model of a probability distribution) that has an infinite number of possible values within any finite segment of its range of values. Continuous distributions form smooth-looking curves. Examples include the normal, student t, and chi squared distributions.

1. Plot discrete probability distributions for simple systems such as flipping coins.

2. Be as one with the law of large numbers.

Average results of many independent observations are stable and predictable

1. Describe the law of small numbers. The tendency to draw unreliable inferences based on a small number of observations.

Relevance: The law of small numbers is a major factor in all kinds of misconceptions and maladies such as hot-hand, lucky charms, and compulsive gambling behavior.

1. Find the mean of a discrete probability distribution. The mean is a weighted average.

μx = ∑ (xi) Pi

Homefun (formative/summative assessment): -- Read section 6.1, Exercises 1, 5, 7 pp. 353 to 355

 Stats Investigation: Simulation of a Random Process ( Teams of two) Purpose: Determine if actual results of flipping a coin match results simulated by humans. Instructions: Each partner in a two person team will record 100 fake coin toss tosses on a sheet of paper using H for heads and T for Tails. The team will then record 200 actual tosses. The experimenters will then circle the runs of 2 or more heads within the data both for the fake and real tosses. For example the following data contains 2 runs of 3, 1 run of 6, and 3 runs of 2 heads: HHHTTHHHTHTHTTTHHHHHHTTTTHHTTTTHHTHH The experimenters will post the number of runs of each length on the board. When all results are posted for the class, the number of runs vs. size of runs is to be plotted for the classes fake and real data ( two separate plots). Questions /Conclusions: Answer with short paragraphs. What is the biggest difference between the real and fake data. Speculate about why this is the way it is. Are humans reliable at generating random numbers? Discuss this both in terms of the law of large numbers and the law of small numbers (p. 392) Resources/Materials: pennies for flipping

 Essential Question: Can standard deviations be added?

Random Variables

1. Calculate the standard deviation of a discrete random variable. (p.410).

Var = ∑ (xi - μx)2 Pi

1. Apply the rules for means.

• if the units of each data point are changed, then the units of the mean are also changed in the same way

• The mean of the sum of random variables is the same as the sum of their means.

1. Apply the rules for the variance of a distribution in which multiple random variables are combined.

• The random variables must be independent

• The variances are additive even when taking the difference between two random variables.

• Standard deviations are not additive.

Homefun (formative/summative assessment): Read 6.2 Exercises 13, 17, 27, 29, 31, 32, 33, 34 pp. 354 to 357

 Essential Question: What will changing the units of measurement do to measures of spread and central tendency?

Linear Transforms

1. What is a linear transform? (See p. 53).

xnew = a + bxold

Example: the linear transform to change from Celsius to Fahrenheit

( ºF ) = 32 + 1.8 ( ºC )

1. State the effect that multiplying each number by a constant and/or adding a constant to each number in a data set has on the following: (This effect is important to know when changing the data points' units.)
 (each data point) + a (each data point) * b mean (mean) + a (mean) * b median (median) + a (median) * b standard deviation no change (std dev) * b IQR no change (IQR) * b range no change (range) * b Q1 and Q3 (Q1) + a & (Q3) + a (Q1) * b & (Q3) * b

Formative assessment: Using the TI-83 calculators and monthly temperature data in Fahrenheit from the Geenville-Spartenburg Airport, calculate and record the mean, median, standard deviation, IQR, range, Q1, and Q3 for both Fahrenheit and Celsius. Convert the recorded Fahrenheit values directly to Celsius. How do the converted values compare to the ones calculated in Celsius.?

Homefun (formative/summative assessment): Exercise 41, 47, 49 pp. 378 to 379

 Essential Question: Do we live in a binary world?
The Binomial Distribution
Relevance: The binomial distribution is frequently used for analyzing and setting up surveys.
1. Be as one with the Binomial Setting, SNIP.
2. Success or failure--observations are divided into these categories.

Number of observations is fixed.

Independent observations

Probability of success is constant

3. Use the binomial coefficient or combinations.

The number of ways of arranging k successes among n observations.

 ( n ) = n! k k! (n - k)!

Note: 0! = 1

1. Calculate binomial probabilities (p. 448).

 P(X=k) = ( n ) pk (1 - p) (n-k) k
1. Calculate combinations and binomial distributions with a TI - 83

• Combinations: no    MATH   PRB  nCr killers

• Binomial Distributions:     2nd     VARS   binompdf or cdf (no, pigs, killed).
 binomepdf : finds the probability for a single value of k. Think p stands for particular. binomecdf : finds the probability for the cumulative values from 0 to k.

Formative assessment: create the 3-coins distribution using the TI-83 calculator.

1. Calculate means and standard deviations for the binomial distr.

(Click here to see an example of count vs fraction type data for the 3 coins example.)

 Data Type Mean Std Dev count np [ np (1 - p) ]^0.5 fraction p [ p (1 - p) / n ]^0.5

Homefun (formative/summative assessment): Read 6.3 Exercises 73, 75, 81, 83, 87, 93 pp. 403 to 405

 Essential Question: When is estimating how many tries before success an issue?

The Geometric Distribution

Relevance: The geometric distribution used for analyzing the probability of an even occurring for the first time, such as the probability of a baseball player getting a hit for the first time vs. the number of times at bat.

1. Be aware of the key differences between binomial and geometric distributions.

Binomial:
Finds the probability that k success will occur in n number of attempts.

Geometric:

Finds the probability that a success will occur for the first time on the nth try.

equation:  P(x=n) = ( 1 - p )n-1 p

1. Be as one with the Geometric Setting on page 464, SPIT
2. Success or failure--observations are divided into these categories.

Probability of success is constant

Independent observations

Trials the number of trials until first success is the goal.

3. Correctly use the geometric distribution for calculating probabilities with the TI-83 calculator.

2nd     VARS   geomepdf or geomecdf (pig, x-ing)

 geomepdf : finds the probability for a single value of x. Think p stands for particular. geomecdf : finds the probability for the cumulative values from 1 to x.

Formative assessment: create the distribution for the probability of hitting the baseball for the 1st time vs number of times at bat using the TI-83 calculator and a batting average of 300 (30% chance of hitting the ball).

Homefun (formative/summative assessment): Exercises 95, 97, 101, 105 pp. 405 to 406

 Essential Question: How can I make an "A" on the test?

Binomial and Geometric Distribution Review

1. Master the vocabulary
2. Work the practice test.
3. Review the objectives.

Homefun (formative/summative assessment): Chapter 6 AP Statistics Practice Test pp.409 to 411

Summative Assessment: Unit Exam objectives 1- 18

Mr

SAM Team--Southside High School's STEM and Computer Science extra-curricular club (Mr. Rogers Sponsor)

Mr. Rogers Teacher's Blog

Mr. Rogers T-shirts

Mr. Rogers Information for Teachers

Mr. Rogers Science Fair Information

Check out other web sites created by Mr. R:

Check out Articles by Mr. Rogers:

Insultingly Stupid Movie Physics is one of the most humorous, entertaining, and readable physics books available, yet is filled with all kinds of useful content and clear explanations for high school, 1st semester college physics students, and film buffs.

It explains all 3 of Newton's laws, the 1st and 2nd laws of thermodynamics, momentum, energy, gravity, circular motion and a host of other topics all through the lens of Hollywood movies using Star Trek and numerous other films.

If you want to learn how to think physics and have a lot of fun in the process, this is the book for you!

 First the web site, now the book!

Mr. Rogers Home | Common Sylabus | AP Comp Sci I | AP Comp Sci II | AP Physics Mech | AP Physics E&M | AP Statistics | Honors Physics|IB Design Tech | Southside

[ Intuitor Home | Physics | Movie Physics | Chess | Forchess | Hex | Intuitor Store |