AP Statistics

Random (simple random)

everyone has an equal chance of being selected

Systematic Random Sample

selected at interval (every nth person)

Sampling w/o replacement

can only be selected once

Stratified Sample Steps

1. divide the population into homogeneous groups (age, gender, race)

2. do an SRS from each group

3. combine into one sample

Cluster Sample Steps

1. divide the population into heterogeneous groups (homeroom)

2. do an SRS of all the clusters

3. survey the clusters selected

multi-stage

draw a sample from a smaller group (10 states, 5 counties per state)

convenience bias

made up of people easy to reach

judgement bias

only choose experts or supporters

size bias

choosing a smaller group to represent a bigger group

response bias

results from survey problems (survey in wrong language)

voluntary response bias

self-selected volunteers

non-response bias

when people of a certain demographic choose not to participate

incorrect response bias

do not answer truthfully

questionnaire bias

question phrasing influences response

block design

the arranging of experimental units into groups(blocks) that are similar to one another, to control for certain factors

matched pairs randomized block design

1. subjects arranged in pairs matched on relevant factors

2. each pair receives both treatments by randomly assigning one treatment to each or each subject could get both treatments

union ( ∪ )

join sets together

list all elements but no repeats

intersection ( ∩ )

what sets have in common

conditional probability

probability of A given B has occured

P(A|B)=P(A∩B) / P(B)

mutually exclusive events

can never occur together, no common outcomes

P(A or B)=P(A)+P(B)

independent events

are not affected by other events

P(A∩B)=P(A)•P(B)

probability of intersection=P(A)•P(B)

mean of random variables

Mx=sum(xi•pi)

standard deviation of random variables

σx=sqrt(Σ(xi-Mi)^2•Pi)

conditions for a binomial setting

1. binary: success or failure

2. same probability: every outcome must have the same chance of occurring

3. independent

4. trials: number of trials set in advance

parameters of a geometric distribution

1. binary: success or failure

2. same probability: every outcome must have the same chance of occurring

3. independent

4. trials: number of trials required to obtain first success

Combination (nCr)

order is not important in arranging groups

Central Limit Theorem

1. If the sample is normal, the sampling distribution will be normal

2. If n>30, the sampling distribution will be normal

The large counts condition

the sample size should be large enough for the distribution of the sample proportion to be approximately normal

n•p>10 and n•(1-p)>10

Conditions to Check for Proportions

1. Random

2. Normal: 10 successes and 10 failures

3. Independent: n<.10N

Conditions to check for Means

1. Random

2. Normal: n>30 or normal sample

3. Independent: n<.10N

normalcdf(L, U, 0, 1)

used to find area under a curve, proportions

plug in z-score (usually in upper)

invNorm(P, 0, 1)

used to find z scores

plug in proportions or probabilities

invT (P, df)

used to find t scores

plug in proportions

when to use a z-score

used only when standard deviation of a population is known and the sample size is greater than 30

when to use a t-score

used when standard deviation of a population is unknown and the sample size is less than 30

p-value

the probability of a result being extreme or more extreme, assuming the null hypothesis is true.

type I error

rejecting the null when the null is true

probability is the same as the significance level

type II error

failing to reject the null when the alternative is true

probability=1-power

power

probability of rejecting Ho correctly

confidence interval conclusion

we are C% confident that the interval from __ to __ captures the (population parameter).

p>alpha

fail to reject Ho, no convincing evidence for Ha

p<alpha

reject Ho, convincing evidence for Ha

chi square conditions

1. Random

2. n<.1N

3. all expected counts>5

chi square goodness of fit (hypotheses and df)

Ho: the distribution is correct

Ha: at least one of the claimed proportions are incorrect

df=# of categories-1

chi square test for homogeneity (how to id, hypotheses, df)

2 separate samples from 2 unique populations

Ho: no difference in distributions of

Ha: difference in distributions of

df=(# of rows-1)(# of columns-1)

chi square test for association/independence (how to id, hypotheses, df)

1 sample from a single population

Ho: no association between

Ha: association between

df=(# of rows-1)(# of columns-1)

expected count for x2 test of homogeneity and independence

(row total•column total)/table total

1 sample z test for population proportions

z=p hat minus the null proportion divided by the square root of the null proportion(1-null proportion) divided by n

1 sample t test for population means

t=sample mean minus null mean divided by standard deviation of the sample divided by the square root of n

2 sample z test for population proportions

Pc or P0 =(x1+x2)/(n1+n2)

2 sample t test for population means

t=x1 minus x2 divided by the square root of s1 squared divided by n1 plus s2 squared divided by n2

interpreting slope on a least squared regression model

as x increases by 1 unit, it is predicted that the y will change by the slope value

t interval for slope of regression (formula and df)

b (your sample slope)+/- [t*(SEb)]

SE is given

df=n-2

use invT for t*

conditions for slope of a regression line

1. sample must be linear

2. random

3. n<.1N

4. no skewed or outlying residuals

5. the standard deviation of y does not vary with x

t test for slope of a regression model

y hat=a+bx

y hat=predicted y value

a=y-intercept

b=slope

Where to find y-intercept when reading a computer output for regression table

Row: Constant

Column: Coef

Where to find slope when reading a computer output for regression table

Row: x variable (under constant)

Column: Coef

Finding r on a computer output model

1. take the square root of R-Sq

2. match the +/- sign to the slope

What is S on a computer output model

standard deviation of the residuals

What is SE Coef on a computer output model

standard deviation of the slope

empirical rule

68% of data is within 1 σ

95% of data is within 2 σ

99.7% of data within 3 σ