AP Stats Section 2 (unit 6-12)

All the mathy things in AP stats

## Experimental Designs

**Completely Randomized****Randomized Block****Matched Pairs**

## Probability Models/Rules

**Two-Way Tables****Venn Diagram****Equally Likely Outcomes:**P(A) = Number of outcomes in event A / Total number of outcomes in sample space

**Complement Rule:**P(A^{c}) = 1 - P(A)**Addition Rule for Mutually Exclusive Events:**P(A or B) = P(A) + P(B)**General Addition Rule:**P(A or B) = P(A) + P(B) - P(A n B)**Conditional Probability:**P(A | B) = P(A n B) / P(B)**General Multiplication Rule:**P(A and B) = P(A n B) = P(A) x P(B | A)**Multiplication Rule for Independent Events:**P(A and B) = P(A n B) = P(A) x P(B)

## Transforming and Combining Random Variables

**Continuous Random Variables:** Can take any value in an interval on the number line.

E(x) = x_{1}p_{1 }+ x_{2}p_{2 }+ x_{3}p_{3}

σ^{2 }= (x_{1 }— μ_{1})^{2}p_{1 }+ (x_{2 }— μ_{2})^{2}p_{2 }+ (x_{3 }— μ_{3})^{2}p_{3}

**Linear Transformations**

**Rule for Means: **μ_{a+bX} = a+b(μ_{x})

**Rule for Variances:** σ^{2}_{a+bX} = b^{2}σ^{2}x

*x*must be a constantadding/subtracting doesn’t change measures of variability (range, IQR, SD), or shape of the probability distribution. It does change measures of center and location (mean, median, quartiles, percentiles).

Multiplying/dividing changes measures of center and location and measures of variability, but not the shape of the distribution.

**Comparing Independent Random Variables**

Where difference is D = X - Y and the sum is S = X + Y

**Rules for combining means:**

μ

_{D }= μ_{X - Y }= μ_{X }- μ_{Y}μ

_{S }= μ_{X + Y }= μ_{X }+ μ_{Y}

**Rules for combining variances:**

σ

^{2}_{D }= σ^{2}_{X - Y }= σ^{2}_{X }+ σ^{2}_{Y}σ

^{2}_{S }= σ^{2}_{X + Y }= σ^{2}_{X }+ σ^{2}_{Y}

Where there is a linear transformation difference is D = bX - cY and the sum is S = bX + c

**Rules for combining means:**

μ

_{D }= μ_{bX - cY }= bμ_{X }- cμ_{Y}μ

_{S }= μ_{bX + cY }= bμ_{X }+ cμ_{Y}

**Rules for combining variances:**

σ

^{2}_{D }= σ^{2}_{bX - cY }= b^{2}σ^{2}_{X }+ c^{2}σ^{2}_{Y}σ

^{2}_{S }= σ^{2}_{bX + cY }= b^{2}σ^{2}_{X }+ c^{2}σ^{2}_{Y}

## Binomial Distributions (6.3)

**Necessary Settings:**

*Binary?*- outcomes can be classified as “success” or “failure”*Independent?*- knowing the outcome of one trial doesn’t tell anything about the outcome of another trial*Number?*- the number of trials of the random process is fixed in advance*Same Probability?*- there is the same probability of success in each trial

**the Binomial Coefficient** - the number of ways to arrange x successes among n trials

nCr

**→**[Math] [Prob] [3]**!****→**[Math] [Prob] [4]

**Precise Functions**

binompdf (n, p, x)

**→**[2^{nd}] [Vars] [A]

**Continuous Functions**

binomcdf (n, p, x)

**→**[2^{nd}] [Vars] [B]P( x < 3 ) = p( x = 0 ) + p( x = 1 ) + p( x = 2 )

expected count of successes:

As the sample size becomes larger, the Binomial Distribution becomes more like a Normal Distribution. If the following conditions are fulfilled then the sample size is large enough to consider the distribution normal:

10% condition

Large Counts Condition: at least 10 expected successes and 10 expected failures

## Geometric Distributions (6.4)

If an experiment has two possible outcomes of *success* (p) and *failure *(1-p), and the trials are independent.

The probability that the first success is on trial number k:

P(

*X*=*k)*= ( 1- p)^{k-1}p

P(X = k)

**→**geometpdf**→**[2nd] [Vars] [F]**→**(prob, x-value)P(X ≤ k)

**→**geometpdf**→**[2nd] [Vars] [G]**→**(prob, x-value)

## Sampling Distributions

Sampling Distributions of p hat

7.2

Sampling Distributions of x bar

7.3

## Confidence Intervals

use t* or z*

Confidence Intervals 1-prop

8.2

Confidence Intervals for p1-p2

8.3

Confidence Intervals for U (mew)

10

confidence interval for a difference between two means (two-samp)

10.2

confidence interval for U diff (paired)

10.2

t-interval for the slope

12.3

## Significance Tests

### About a Proportion - 1-prop (9.2)

**One sided**

**S:** H_{0 }: p = p_{0}

H_{a }: p > p_{0 }-or- H_{a }: p < p_{0}

**P: **One sided z-test for p̂

Random: The data comes from a random sample from the population of interest.

10%: when sampling without replacement,

*n*< 0.1NLarge Counts: Both

*np*_{0 }and n( 1 -*p*_{0}) are at least 10

**D:**

P(p < p_{0}) **→**normcdf** → **[Stat] [tests] [5] **→ **(-1e99, z, 0, 1)

P(p > p_{0}) **→**normcdf** → **[Stat] [tests] [5] **→ **(z, 1e99, 0, 1)

**C:** Because our p-value of *p* < *a* = 0.05, we reject H_{0} . We have convincing evidence that…

**Two sided**

**S:** H_{0 }: p = p_{0}

H_{a }: p ≠ p_{0 }

**P: **Two sided z-test for p̂

**D: **P(p ≠ p_{0}) **→**normcdf** → **[Stat] [tests] [5] **→ **(-1e99, z, 0, 1)x2

** interpret the p-value - **“assuming H

_{0}is true, there is a

*p*probability of getting a sample mean of ___ by chance alone in a random sample of n.”

### Power

the probability that the test will find convincing evidence that H_{a} is true when H_{0} is true.

** interpret - **“If the true proportion of

*population*

**is p**

_{0}, there is a

*power*probability of finding convincing evidence that H

_{a }: p < p

_{0 }is true.”

Power increases when:

the sample size n is larger

the significance level a is larger

the null and alternative parameters are further apart

## About a Difference in Proportions - 2-prop (9.3)

significance test about a mean

11

significance test with a difference in means (2-samp)

11.2

significance test about U diff (paired t-test)

11.2

significance test for the slope

12.3

## X^{2} Test for Goodness of Fit (12)

12

test for homogeneity

11.2

o

test for independence/association

11.2

# AP Stats Section 2 (unit 6-12)

All the mathy things in AP stats

## Experimental Designs

**Completely Randomized****Randomized Block****Matched Pairs**

## Probability Models/Rules

**Two-Way Tables****Venn Diagram****Equally Likely Outcomes:**P(A) = Number of outcomes in event A / Total number of outcomes in sample space

**Complement Rule:**P(A^{c}) = 1 - P(A)**Addition Rule for Mutually Exclusive Events:**P(A or B) = P(A) + P(B)**General Addition Rule:**P(A or B) = P(A) + P(B) - P(A n B)**Conditional Probability:**P(A | B) = P(A n B) / P(B)**General Multiplication Rule:**P(A and B) = P(A n B) = P(A) x P(B | A)**Multiplication Rule for Independent Events:**P(A and B) = P(A n B) = P(A) x P(B)

## Transforming and Combining Random Variables

**Continuous Random Variables:** Can take any value in an interval on the number line.

E(x) = x_{1}p_{1 }+ x_{2}p_{2 }+ x_{3}p_{3}

σ^{2 }= (x_{1 }— μ_{1})^{2}p_{1 }+ (x_{2 }— μ_{2})^{2}p_{2 }+ (x_{3 }— μ_{3})^{2}p_{3}

**Linear Transformations**

**Rule for Means: **μ_{a+bX} = a+b(μ_{x})

**Rule for Variances:** σ^{2}_{a+bX} = b^{2}σ^{2}x

*x*must be a constantadding/subtracting doesn’t change measures of variability (range, IQR, SD), or shape of the probability distribution. It does change measures of center and location (mean, median, quartiles, percentiles).

Multiplying/dividing changes measures of center and location and measures of variability, but not the shape of the distribution.

**Comparing Independent Random Variables**

Where difference is D = X - Y and the sum is S = X + Y

**Rules for combining means:**

μ

_{D }= μ_{X - Y }= μ_{X }- μ_{Y}μ

_{S }= μ_{X + Y }= μ_{X }+ μ_{Y}

**Rules for combining variances:**

σ

^{2}_{D }= σ^{2}_{X - Y }= σ^{2}_{X }+ σ^{2}_{Y}σ

^{2}_{S }= σ^{2}_{X + Y }= σ^{2}_{X }+ σ^{2}_{Y}

Where there is a linear transformation difference is D = bX - cY and the sum is S = bX + c

**Rules for combining means:**

μ

_{D }= μ_{bX - cY }= bμ_{X }- cμ_{Y}μ

_{S }= μ_{bX + cY }= bμ_{X }+ cμ_{Y}

**Rules for combining variances:**

σ

^{2}_{D }= σ^{2}_{bX - cY }= b^{2}σ^{2}_{X }+ c^{2}σ^{2}_{Y}σ

^{2}_{S }= σ^{2}_{bX + cY }= b^{2}σ^{2}_{X }+ c^{2}σ^{2}_{Y}

## Binomial Distributions (6.3)

**Necessary Settings:**

*Binary?*- outcomes can be classified as “success” or “failure”*Independent?*- knowing the outcome of one trial doesn’t tell anything about the outcome of another trial*Number?*- the number of trials of the random process is fixed in advance*Same Probability?*- there is the same probability of success in each trial

**the Binomial Coefficient** - the number of ways to arrange x successes among n trials

nCr

**→**[Math] [Prob] [3]**!****→**[Math] [Prob] [4]

**Precise Functions**

binompdf (n, p, x)

**→**[2^{nd}] [Vars] [A]

**Continuous Functions**

binomcdf (n, p, x)

**→**[2^{nd}] [Vars] [B]P( x < 3 ) = p( x = 0 ) + p( x = 1 ) + p( x = 2 )

expected count of successes:

As the sample size becomes larger, the Binomial Distribution becomes more like a Normal Distribution. If the following conditions are fulfilled then the sample size is large enough to consider the distribution normal:

10% condition

Large Counts Condition: at least 10 expected successes and 10 expected failures

## Geometric Distributions (6.4)

If an experiment has two possible outcomes of *success* (p) and *failure *(1-p), and the trials are independent.

The probability that the first success is on trial number k:

P(

*X*=*k)*= ( 1- p)^{k-1}p

P(X = k)

**→**geometpdf**→**[2nd] [Vars] [F]**→**(prob, x-value)P(X ≤ k)

**→**geometpdf**→**[2nd] [Vars] [G]**→**(prob, x-value)

## Sampling Distributions

Sampling Distributions of p hat

7.2

Sampling Distributions of x bar

7.3

## Confidence Intervals

use t* or z*

Confidence Intervals 1-prop

8.2

Confidence Intervals for p1-p2

8.3

Confidence Intervals for U (mew)

10

confidence interval for a difference between two means (two-samp)

10.2

confidence interval for U diff (paired)

10.2

t-interval for the slope

12.3

## Significance Tests

### About a Proportion - 1-prop (9.2)

**One sided**

**S:** H_{0 }: p = p_{0}

H_{a }: p > p_{0 }-or- H_{a }: p < p_{0}

**P: **One sided z-test for p̂

Random: The data comes from a random sample from the population of interest.

10%: when sampling without replacement,

*n*< 0.1NLarge Counts: Both

*np*_{0 }and n( 1 -*p*_{0}) are at least 10

**D:**

P(p < p_{0}) **→**normcdf** → **[Stat] [tests] [5] **→ **(-1e99, z, 0, 1)

P(p > p_{0}) **→**normcdf** → **[Stat] [tests] [5] **→ **(z, 1e99, 0, 1)

**C:** Because our p-value of *p* < *a* = 0.05, we reject H_{0} . We have convincing evidence that…

**Two sided**

**S:** H_{0 }: p = p_{0}

H_{a }: p ≠ p_{0 }

**P: **Two sided z-test for p̂

**D: **P(p ≠ p_{0}) **→**normcdf** → **[Stat] [tests] [5] **→ **(-1e99, z, 0, 1)x2

** interpret the p-value - **“assuming H

_{0}is true, there is a

*p*probability of getting a sample mean of ___ by chance alone in a random sample of n.”

### Power

the probability that the test will find convincing evidence that H_{a} is true when H_{0} is true.

** interpret - **“If the true proportion of

*population*

**is p**

_{0}, there is a

*power*probability of finding convincing evidence that H

_{a }: p < p

_{0 }is true.”

Power increases when:

the sample size n is larger

the significance level a is larger

the null and alternative parameters are further apart

## About a Difference in Proportions - 2-prop (9.3)

significance test about a mean

11

significance test with a difference in means (2-samp)

11.2

significance test about U diff (paired t-test)

11.2

significance test for the slope

12.3

## X^{2} Test for Goodness of Fit (12)

12

test for homogeneity

11.2

o

test for independence/association

11.2