AP stats

holy cram

Quantitative variable

takes numerical values for a measured of counted quantity

categorical variable

takes on values that are category names or group labels

• bar groups show frequency (how many) or relative frequency (percent)

misleading graphs

• vertical axis must start at 0

• beware of using images for bar graphs

segmented bar graph

stack up bars to make 100%

mosaic plot

segmented bar graph where the width of bars is proportional to the group size

association

if knowing the value of one variable helps us predict the other variable

• segmented bar graphs are different

2 different quantitative data

discrete and continuous

discrete quantitative data

countable number of values

continuous quantitative data

infinite values

SOCV

shape, outliers, center , variability with context]

interpret standard deviation

“The context typically varies by SD from the mean of “

how to find variance?

(SD)²

Greated affected by outliers (nonresistant)

mean and standard deviation

resistant to change

median

1.5 x IQR method

low outlier < Q₁ - 1.5(IQR)

high outlier > Q₃ + 1.5(IQR)

SD Method for outliers

low outlier < mean - 2(SD)

high outlier > mean + 2(SD)

calculate IQR

MAX - MIN

Boxplots

5 number summary:

min, Q₁ , med, Q₃, max

Interpret percentile

“The p^th percentile is the value that has the p% of the data less than or equal to it.”

Q1 percentile

25th percentile (0.25)

What percentile is the Median?

50th percentile (0.50).

Q₃ percentile

75th percentile (0.75)

interpret Z-score

“context” is z-score standard deviations above/below the mean of “

what do z-scores show?

They show position relative to other values in the distribution

Empirical rule (68-95-99.7)

how to find z-score for a given proportion

use Table A, or TI84: InvNorm

Describe a relationship (DUFS + Context0

Direction (positive/negative, none)

Unusual features (outliers, clusters)

Form (linear or nonlinear)

Strength

Interpret Correlation ( r )

“The linear relationship between x and y is strength and direction .”

interpret coefficient of determination r²

“the percent of the variation in y explained by the linear relationship with x”

does correlation equal causation?

NO

how to calculate residual

residual = Actual - Predicted

(r=a-p)I

Interpret residual

“the actual context was residual value above/below the predicted value for x=#”

interpret y-intercept

“when x=0 context, the predicted y-context is y-intercept.”

interpret slope

“For each additional x-context the predicted y-context increases/decreases by slope.”

Least Squares regression line

minimizes the sum of the squared residuals

horizontal outliers

tilt the least squares regression line

vertical outliers

shift the least squares regression line up or down

high leverage outliers

very large of small x-values

influential outliers

if removed, big changes happen to the slope, y-int and correlation ( r ).

choosing best regression model

1. check the scatter plot for a linear pattern

2. check residual plot for no leftover pattern

3. check for the r² that is closest to 1.

convenience sample

people are easy to reach, can lead to bias

voluntary response

people choose to respond, can lead to bias

what samples can lead to bias?

convenience sample and voluntary response

simple random sample (SRS)

1. label individuals

2. randomize

3. select

label individuals in SRS

assign numbers or write names on slips of papers

randomize in SRS

random number generator (no repeats) or names in a hat (shuffle)

stratified random sample

splot the population into groups (strata) then choose an SRS from each strata.

Homogenous grouping

each strata has individuals with shared attributes or characteristics

what leads to the best estimates in a sampling method?

low bias and low variability

cluster sample

heterogenous groups, sample all from some groups

systematic random sampling

choose a random starting point and go from equal intervals

undercoverage

some people are less likely to be chose

ex. calling landlines, surveying homeowners

nonresponse

people cant be reached or refuse to answer

ex. don’t answer of Hang up phone calls

response bias

problems in the data gathering instrument or process

ex. people lie (self reported responses), wording of quesion

observational study

no treatment imposed

experiment

treatment imposed, allow us to show causation

well-designed experiment steps

1. comparison (2 or more treatments)

2. random assignment

3. replication (more than 1 in each treatment group)

4. control (keep other variables constant)

what does random assignment do?

allows us to show causation

placebo effect

when a fake treatment works

blinding

when subjects (single blind) and/or experimenters (double blind) don’t know about treatments

randomized block design

separate subjects into blocks and then randomly assign treatments within each block

block

group of experimental units that are similar

matched pairs design

subjects are paired (block size 2) and then randomly assigned to a treatment and each subject receives two treatments

statistically significant

when results of an experiment are unlikely (less than 5% (0.05)) to happen purely by chance.

If statistically significant, we have convincing evidence the treatment caused the difference.

what does a random sample allow us to do?

Allows us to generalize our conclusions to the population from which we sampled

long run relative frequency

• always between 0 and 1 (inclusive)

  • predictable

short run relative frequency

predictable

law of large numbers

simulated probabilities tend to get closer to the true probability as the number of trials increase.

sample space, list of all possible outcomes

P(E) = # outcomes in E/ total # outcomes in sample space

complement rule, probability of an event not happening

P(A^c) = 1- P(A)

P(A and B)

P(A n B) where both occur

P(A or B)

P(A U B) one of the other or both

addition rule in probability

P(A or B) = P(A) + P(B) - P(A and B)M

Mutually exclusive

Events A and B can’t occur together

P(A and B) = 0, so P(A or B) = P(A) + P(B)co

conditional probability

the probability of one event given another has occurred

P(A | B)= P(A and B) / P(B)

independent events

knowing whether or not one event occurs does not changed the probability of the other event

If P(A) = P(A | B) = P(A | B^c)

then A and B are independent

general multiplication rule

If A and B are independent

P( A and B) = P(A) x P(B)

probability of getting at least 1

1 - P(none)

discrete random variable

takes a countable number of values with gaps between

continuous random variable

has infinite values with no gaps

ex. uniform, normal

sample size

as sample size increases, variability decreases

central limit theorem

the sampling distribution of x-bar is approximately normal when the sample size is => 30

confidence interval

point estimate +- margin of error

P.E = A+B/2

M.E = B - A / 2

interpret confidence interval

we are % confident that the interval from A to B captures the true context

• all values between A and B are plausible

Margin of Error

increased confidence, increased M.E leading to a wider interval

increased sample size, decreased M.E leading to narrower interval

interpret confidence level

if we take many, many samples and calculate a confidence interval for each, about % if them will capture the true context.

conditions for constructing a confidence interval for proportion

1. random condition (must have random sample)

2. 10% condition (when sampling w/o replacement, check n <= 10% (N population))

3. Large Counts condition ( n(p-hat) >= 10 and n( 1-(p-hat) ) => 10

interpret p value

assuming the null hypothesis is true ( p = ho context), there is a p-value probability of getting a p-hat of or more extreme purely by chance.

conclusion

Because p-value < significance level (or p-value > sig. lvl.), we reject the null hypothesis (or fail to reject) and we do (or do not) have convincing evidence for Ha context.

type I error

the null hypothesis (context) is true, but we find convincing evidence for Ha(context

type II error

the ha(context) is true but we don’t find convincing evidence for Ha(context)