2nd VARS
normalcdf
enter (lower, upper, mean, standard deviation) to calculate the cumulative probability for a normal distribution.
Gives you percentage in decimal form (0.0668 = 6.68%)

Normal Curve

symmetrical about the mean = mode = median
never touches x-axisthe
total area under curve = 1
the shape of the graph is influenced by
- mean (horizontal) shift right
- standard deviation (width of the graph)

Bimodal Distributions

~ a type of discrete probability ~

Bernoulli sequence: a sequence of hits that have only 2 outcomes (coin flip)

identical trails
must be independent

If x is a random varible with bimodal distrubtions

Probability of x no of successes in n trails

Order is not important

Mean & Standard Deviation

The values of μ & p change the graph shape

μ: as μ increases, more lines make the graph closer to a bell-shaped curve

p: as p increases (consent), the graph tail is strengthened more to the left

if p < 0.5 tail on the right (very skewed)

A response variable measures an outcome of a study (y) (explanatory predicts)
An explanatory variable may help or explain the change in a response variable (x)

Scatterplot: shows the relationship between 2 quantitative variables measured for the same individual

each individual appears as a point

Correlation: r gives the direction and measures the strength of the linear association between 2 quantitative variables

if asked for r, and you only have r² square root to find r
The Linear association between x and y is _____ and ______.

Regression Line: models show how a response variable (y) changes as an explanatory variable (x) change

y = a +bx

Residuals: distance between actual and predicted

actual y - predicted

y= residual

The actual y was redisdual above/below the predicted y of x-context.

High Leverage Points

influential points have the most significant impact
an outlier can also be a high-leverage point
- not all outliers are high-leverage
away from data but with the same x-values
not all outliers are influential
- they can be influential though
if both it can be an outlier

To determine if an outlier is considered:

leverage
x-values
residual distance from the regression line

Sx of the residuals: measures the size of a typical residual

The actual y is typically about Sx away from the predicted y by the LSRL

lethe ast square regression line

r²% of the variations of x-context is explained by the linear relationship with log(y-context)

When a scatterplot shows a curved relationship between 2 quantitative variables, transform one or bothvariablesse to crea linear associations

choose a model whose residual plot has the most random scatter (no curve)
if more than one model produces a randomly scattered residual plot, choose the model with the largest coefficient of determination (r²/r)

Simple Random Sample

relies on using a selection method that provides each survey participant with an equal chance of being selected
based on probability and random selection
more likely to be representative of the total population (free of bias)

Label (give each individual a number)
randomize (use a number generator)
select (choose individuals that correspond)

Calculator

math
prob
randInt(up,lower,h)

For a population of 30 students, choose 5 random students

Choose SRS with a table

label
randomize
select

more Biased

Stratified Random Sampling

for surveys (total population is divided into groups (strata)
grouping of similarities (SRS from each group)
used by researchers when trying to evaluate data

define the groups
define the sample size (ratio)
randomly select from each group
review results

Low Bias & Low Variability

Low bias: your data is standard (symmetric)

Low variability: your data is similar to the mean

If your sample has both it is most likely the population and generalizations to the population are most likely to be accurate

Advantages

symmetric demographics
fair method
helps efficient
accurate data

Disadvantages

prior knowledge
may not be representative
more complicated
selection bias

Strata: groups of individuals in a population

a strata random sample is selected by choosing an SRS from each strata

Sampling variability: the static information from a sample will vary as the random sampling is repeated

will decrease as the sample size increases

cluster: a group of individuals that are located near each other

divides the population into groups or clusters
- selected to make up your total sample group for a study
useful when surveying a large population and natural grouping
- might be bias

define the population and cluster size
generate your clusters
randomly select clusters
collect data
analyze and interpret data

Advantages

cost-effective
efficiency
speed natural grouping

Disadvantages

bias on sampling
complexity

Groups are not similar

sample all from some groups

Systemic random sampling is a large population selected, according to a random starting point but with a fixed periodic interval

calculated by dividing the population size by the desired sample size

confirm population total
determine sample size
determine sampling interval
select a random stat point
add sampling interval until the desired sample

	Cluster sampling	systemic sampling	stratified sampling	simple random sampling
Population	The population is divided into clusters/groups	population is divided into groups	The population is divided into strats or subgroups	while the population is considered
sampling unit	clusters are selected randomly but the entire population of clusters is surveyed	every nth unit in the population is selected for surveying	Individuals within each are randomly chosen for surveying	Individuals are randomly selected from the population for surveying
homogeneity with the sample unit	high homogeneity within each selected cluster	assumes homogeneity within selected intervals	lower homogeneity with each strat/subgroup	assumed homogeneity across the entire population
complexity	fewer stages of the sampling method involved	simple to implement with one-stage sampling	more stages of sampling involved	simple to add with 1-stage sampling

Undercoverage occurs when some members of the population are least likely to be chosen or cannot be chosen in a sample

a survey of households (excludes homeless people, prisoners, students, dormitories.)

Response bias occurs when there is a systemic pattern of inaccurate answers to a survey question.

Confounding occurs when 2 variables are associated in a way that their effects on a response variable cannot be distinguished from each other

possible different variable (3rd variable)

Treatment is a condition applied to individuals in an experimentA

A placebo is a treatment that has no active ingredient

Factor is an explanatory variable that is manipulated and may cause a change in the response variable

the different values of a factor are called levels

A control group is used to provide a baseline for sampling the effects of other treatments

2 different groups, 1 with treatment and 1 without (control variable)

The placebo Effect describes the fact that some subjects will respond favorably to any treatment..nt

Single-blind: either subject or people who interact with them and measure the response variable don’t know which treatment a subject is receiving

neither subject nor people (double Blind)

Block is a group of experimental units

randomized block is carried out within each block

Matched Pairs Design is a common experimental design for comparing 2 treatments tuseses a block of size 2.

Replication of Trials

Observation: select people
Experiment: random assignment, treatment, causation
- 1. comparison
- random assignment
- control
- replication

Randomized block design

you divide your participants into subgroups
- within the blocks they have similarities

Statistical significance

helps quantify results likely to be based on factors of interest
- (chance or not?) (lucky or unlucky)

Scope of inference

random selection
random assignment
- allows for inference from population (cause/effect)

A random sample will enable us to generalize our conclusions to the population from which we have sampled

when can we decide on the causation
- when we generalize

Probability

any outcome of a random process is a number between 0 and 1, that describes the proportion of times that outcome would occur in a very long series of trials.
an outcome that never occurs has a probability of zero
an outcome that appears/happens on every trial has a probability of 1
an outcome that happens 1.2 of the time has a probability of 1.2

The law of large numbers says that if we observe more and more trials of any random process, the proportion of times that a specific outcome occurs approaches its probability.

After many many contexts, the proportion of times that context A will occur is about P(A).

Simulation

describe how to set up (use a random process)
identify what your recording
perform many trials
use results to answer question

A sample size is the set of all possible outcomes of an experiment or simulation..on

Mutually Exclusive Events

2 or more events that cannot occur at the same time
- disjoint event (P(A or B) = P(A) + P(B)

RULES

between zero and one
all outcomes probability = one
probability of an event is one minus the probability

Venn diagrams

represented by 2 circles that overlap to show a relationship

General Addition Rule

if A and B are 2 events resulting from the same random process

Introspection (n) (and)

all outcomes that are common to both sides

Union(U)(or)

all outcomes that are not common to both sides

Conditional probability

one event happens given that another event is known to have happened

A tree diagram shows the sample space of a random process including multiple stages.

and calculates probability

A random variable takes numerical values that describe the outcomes of a random process.

A probability distribution gives the possible values and their possibilities

to be a valid probability model
- between zero and one
- sum of all probability = 100%

Discrete random variable

a countable set of possible variables with gapes on a number line (whole number only)

Histogram of probability distribution (no gaps)

values of a random variable
probability
one bar per each x-value

The mean or expected value is its average over many trials of the same random process

to find multiple each possible by its probability then add to the sum

If many many context are randomly selected by the average amount of context of random variable would be about _____ (units).

can also be found from a cumulative probability distribution for the random variable

The median of a discrete random variable is the midpoint of a distribution that varies from the .mean.

If many contexts are randomly selected, the context will typically vary from the mean of x by about standard deviation (units).

Transforming probility distributions probability

adding/subtracting
- measures of center (mean or median)
- doesn’t change variability
- doesn’t change the shape of the probability distribution
multiplying/dividing
- measures the center by b
- measures the variability by b
- doesn’t change the shape of the distribution

Effect of a linear transformation on random variable