Statistics

Probability and Simulation

Permutation & Combination

Addition and Multiplication Principle

Permutation

a permutation is the arrangement of objects taken from a ser

The ordir of the objects is important

if there are n distincts objects. we have n! ways of arranging all the objects in a row

if we choose r objects from n distinct objects, wer have nPr ways to arranging the r objects

if some of the objects are identical, the no. of permutations will be less

if there are n objects in total with p identical objects, we have n!/p! ways of arranging all the objects in a row

Conbination

order is not important

there are c ways of selecting r objects

nCr

Circular permutation

Subtopic

each object has two "neighbors" when arranged in a circle

we have (n-1)! wys to arrange n distinct objects in a circle

Simulation

The Idea of Probability

chance behavior is unpredicatable in the short run but has a regular and predictable pattern in the long run

random

individual outcomes are uncertain

a regular distribution of outcomes in a large no. of repetitions

probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very large series of reptitions

Probability Models

Sample space

S

the set of all possible outcomes

event

any outcome or a set of outcomes of a random phenomenon

probility modal

a mathematical description of a random phenomenon consisting of two parts

a sample space S

a way of assigning probabilities to event

Probility of an Event with Equal Likely Outcomes

Subtopic

P(A) = n(A)/s(A)

Important Probability Results

0≤P(A)≤1

P(A) =1, event A will certainly occur

P(A)= 0, event A will never occur

P(S) = 1

complement rule

Probability("not" an event) = 1 - Probability(event)

two events A and B are disjoint

mutually exclusive

indenpendent event

chance of one event happen or does not happen doesn't change the probability that the other event occurs

P(A and B) = P(A) * P(B)

General Probability Rules

Conditional probability

Random variables

random varible X

A numerical value assigned to an outcome of a random phenomenon.

Discrete and Continuous Random Variables

discrete random variable

random variable with a countable number of outcomes

Probability distribution

a list of the possible values of the DRV together with their respective probabilities

continuous random variable

random variable that assumes values associated with one or more intervals on the number line

Means and Variances of Random Variables

Probility distribution

p(xi) = P(X=xi)

Condition

Probability Density Function

The Binomial and Geometric Distributions

The Binomial Distributions

Subtopic

Bernoulli distribution

Two possible outcomes: "success" or "failure"

p=P(success)

q=P(failure)=1-p

p+q=1

Binomail Distribution

Two outcomes

The experiment is repeated a number of times independently

Conditions

only 2 outcomes in each trial

a fixed number n of trials

trials are independent

probility of success for each trial is the same

Binomial Formula

X be the number of success after n trials

X is binomially distributed with parameters n and p

X~B(n,p)

Binomial distribution is very important in statistics when we wish to make inferences about the proportion p of "success" in a popultaion

Binomial Probability

Probability distribution function

pdf assins a probability to each value of X

Cumulative distribution function

cdf of X calculates the sum of probabilities for 0, 1, 2......, up to the value X

Normal Approximation to Bionomial Distribution

The formula for binomial probabilities becomes awkward as the no. of trials n increase

When n is large, the distribution of X is approximately Normal

N(np, √np(1-p)

The accuracy of the Normal approximation improves as the sample size n increases

The Geometric Distributions

Geometric Distribution

In a geometric random variable. X counts the number of trals until an event of interest happens

Conditions

Each observation falls into one of just two categories, "success" or "failure"

The observations are all independent

The probability of a success, p is the same for each observation

The variable of interest, X, is the number of trials required to obtain the first success

Calculating Geometric Probabilities

P(X=n) = (1-p)^(n-1)p

Geometric Mean and Standard Deviation

Mean = 1/p

Varience = std^2 = (1-P)/p^2

Sampling Distributions

Sampling Distributions

Parameter & Statistic

A parameter is a number that describes the population

A statistic is a number that can e computed from the sample data without making use of any unknown parameters

Statistics comes from samples while parameters come from populations

Sampling Variability

Different samples will give different values of sample mean and proportion

The value of a statistic varies in repeated random sampling

Sampling Distribution

the distribution of values taken by the statistic in all possible samples of the same size from the same population

Bias and Variability

To estimate a parameter is unbiased

The mean of its sampling distribution is equal to the true value of parameter being estimated

The statistic is called an unbiased estimate of the parameter

Variability

described by the spread of sampling distribution

determined by the sampling design and size of the sample

larger samples give smaller spread

Sample Proportions

The mean of sampling distribution of p is given by p

The standard deviation of the sampling disribution of p is given by √(p(1-p)/n)

Only used when the population is at least 10 times as large as the sample

a Normal approximation

np≥10

n(1-p)≥10

Sample Means

mean of sampling distribution = µ

standard deviation of sampling distribution = std/n

Central Limit Theorem

for large sample size n, the sampling distribution os x is approximately Normal for any population with finite std

the mean and standard deviation of the sampling distribution is given by µ and std/√n

Estimating with confidence

Confidence Interval

Introduction

Statistical inference

Provides methods fro drawing conclusions about a population from sample data

Estimation

Porcess of determining the value of a population parameter from information provided by a sample statistic

point estimate

a single value that has been calculated to estimate the unknwn parameter

Confidence interval

a range of plausible values that is likely to contain the unknown population parameter

range of values

Subtopic

generated using a set of sample data

center

sample statistics

Confidence Interval and Confidence Level

for a parameter

calculated from the sample data: statistic ± margin of error

gives the probability that the interval will capture the true parameter value in repeated samples

Margin of error

The range of values to the left and right of the point estimate in which the parameter likely lies

Standard error

the standard error of a reported proportion or percentage p measures its accuracy

the estimated standard deviation of that percentage

std/√n

z*std/√n

Confidence interval for a Population Mean

Conditions

sample taken from SRS

Normality: n is at least 30

Independence : population size is at least 10 tmes as large as the sample size

General Procedure for Inference with confidence Interval

Step 1 : state the parameter of interest

Step 2: Name the inference prodedure and check conditions

Step 3: calculate the confidence interval

Step 4: Interpret results in the context of the problem

How Confidence Interval Bheave and Determining Sample Size

smaller margin of error

The confidence level C decreases

the population standard deviation decrease

the sample size increase

Sample size

Extimating a Population Mean

Std of population unknown

Conditions:

Samples takne from SRS

Population have a normal distribution

Individual observations are dndependent

Standard error of sample mean

The t Distribution

Not normal, it is a t distribution

degree of freedom

df = n-1

because we are using the sample standard deviation s in our calulation

write the t distribution with k degrees of freedom as t(k)

Density curves

similar in shape to the standard Normal curve

The spread of the t distribution is a bit greater than that of the standard Normal distribution

as k increase. the denstiy curve approaches the N(0,1)curve ever more closely

The One-Sample t confidence Intervals

unknown mean µ

a level C confidence interval for µ is: x ± t*s/√n

t* is the critical value for the t(n-1)

The interval is exactly correct when the population distribution is Normal

The interval approximately correct fro large n in other cases

Paried t Procedures

compare the responses to the two treatments on the same sjubects, applys one sample t procedure

matched pairs design

before-and-after measurement

The parameter µ is a paired t procedure is

the mean difference in the responses to the two treatments within matched pairs of subjects in the entire population

the mean difference in response to the two treatments fro individuals in the population

the mean difference between before-and-after measurements for all individuals in the population

Robustness of t Procedures

robust against non-Normality of the population

not robuts against outliers

Using the t procedures

more important than the assumption that the population distribution is Normal

Sample size less than 15, data close to normal

large samples

Estimating a Population Proportion

Conditions for Inference about a proportion

level C confidence for population proportion: p±z*√(p(1-p)/n)

Conditions:

The datas are taken from SRS

Normality

np≥10

n(1-p)≥10

Individual observations are dndependent

Choosing the sample size

m = z*√(p(1-p)/n)

n=(z*/m)^2 p(1-p)

Testing a Claim

The basics

using a confidence interval to estimate the population parameter

using a statistical test to asses the evidence provided by data about some claim concerning a populatin

Basic ideas

An outcome that would arely happen if a claim were true is good evidence that the claim is not true

How rare is rare

Stating Hypohteses

null hypotheses

the statement being tested in a significance test

H0 : μ1 = μ2

alternative hypotheses

the claim about the population that we are trying to find evidence for

H0 : μ1≠μ2

P-value

The probability of a result at least as far out as the result we actually got

Significant test

Signficance tests are performed to determine whether the ovserved of a sample statistic differs significant from the hypothesized value of a population parameter

porcedure

Step1: Hypothesis

Identity the population of interest and the parameter

State hypotheses

Step 2: Conditions: Choose the appropriate inference procedure

z-Test for population mean

population std known

z = (x-µ)/(std/√(n))

t-Test for population mean

population std unknown

t = (x-µ)/(s/√(n))

Sample from SRS

Samples indenpendent from each other

Sample follow Noraml distribution

Step3: Calculations

Calculate the test statistic

Find the P-value

by GC

value used

µ>µo, p

µ<µo, p

µ≠µo, 2p

Step 4: Interpretation

Interpret the P-value or make a decision about Ho using statistical significance

Conclusion, connection and context

Test from confidene interval

intimate connection between confidence and significance and significance

if we are 95% confident that the true µ kues in the interval, we are also confident that the values of µ that fall outside ourinterval are incompatible with the data

Confidence intervals and Two-Sided Test

a two-sided significance test rejects a hypothesis exactly when the value µo falls outside a level 1- alpha confidence interval for µ

The link between two-sided significance tests and confidence interval is called duality

in two-sided hypothesis test, a significance test and a confidence interval will yield the same conclusion

cannout use a confidence interval in place of a significance test for one-sided tests

Use and Abuse of Tests

valued because an effect that is ulikely to occur simply by chance

choosing a Level of significance

How plausible is Ho

What are the consequedces of rejecting Ho

no sharp border between statistically significan and tattistically insignificant

increasing strong evidence as the P-value decrease

Statistical Significance and Practical Importance

very small effects can be highly significan

A statistically significant effect need not be parctically important

Statistical inference is not valid for all sets of data

Beware of multile analyses

Type I and Type II error

If we reject Ho when Ho is actually true, we have a committed type I error

The significance level of any fixed level test is the probability of type I error

If we fail to reject Ho when Ho is false, we have committed a type II error

power test

when a particular alternative value of parameter is true is called the power of the test againgst that alternative

The power of a test against any alternative is 1 minus the probability of a type II error for that alternative

1-ß

Increase the power of a test

increase alpha

consider a particular alternative that is farther away from µ

Increase the sample size

Decrease std through improving the measurement process and restricting attention to a subpopulation

Significance Tests in Pratice

The one sample t-test

The One-propottion z test

z = P-Po/√(Po(1-Po)/n)

Normality condition

nPo≥10

n(1-Po)≥10

Comparing Two population Parameters

Population std known

Two-Sample z Statistic

Population std unknown

The Two-Sample t procedure

level C confidence interval

two-sample t statistic

Degree of Freedom

Choose the small one

Two sample tests about a population proportion

Conditions:

SRS

Indenpendence

Normality

Inference for Distribution of Catagorical Variables

The Chi-Sqaures Test fro Goodness of Fit

P-value and significant test

The Chi-Sqaure Distribution

alternative hypothesis: at least one catagory data is different the null hypothesis

df = no. of catagoris-1

The Chi-Square Test for homogeneity of populations

expected count = (row total*column total)/n

Conditions:

SRS

Independence

Normality

Inference for Regression

Exploring Data

Displaying Distributions with Graphs

Graphs for Categorical Data

Dotplots

Pie Chart

Bar Chart

Graphs for Quantitative Data

Histrogram

Stemplots

Cumulative Frequency Plots

Time Plots

Describing a Graphical Display

Mode

The mode is one of the major "peaks" in a distribution

Unimodal

distribution with exactly one mode

bimodal

Distribution with 2 mode

Center

The center which roughly separates the values roughly in hafl

Spread

The scope of the valves from the smallest to the largest values

Clusters

The natural subgroups in whch the values fall into

gaps

Holes in which no values fall into

Outliers

Extreme values in the distribution

Due to errors

Requires futher analysis

Due to natural variation in the data

Shape of Graphical Displays

Skewed

Describing Distributions with Numbers

Mearuring the Center

Mean

The mean of a series of variables is the arithmetical average of those numbers

Median

The Median is the middle number of a set of a data arranged in nmbrical ascending/descending order

NO. of items is ole

E.g: {3.5,9,15,15},Median=9

NO. of items is ever

E.g:{3,5,9,10,15,15} Median = (9+10)/2

Reisitant Measure

iThe median value is not affected by outlier values. Wd describe the median as a resistant measure.

Measuring Spread

Range:

difference between large and small observation

Percentile

Median, M, the 50 percentile

Q1, the 25 percentile

Q3, the 75 percentile

Interquartile range(IQR)

The difference between the first and third quartile

Box Plot

Change Unit of Measures

Comparing Distribution

Describing Location in a Distribution

Mearures of Relative Standing and Density Curves

Position of the Score(How your score compares to other peoples's score)

Z-Score

To measred how many standard deviations is avay from the mean

Percentile

The pth percentil of a distribution is defined as the value with p percent of the ovservation less or equal to it

Chebyshey's Inequality

In any distribution, the percentage of observations falling within standard deviation of the mean is at least

Density Curves

A mathematical model for the distribution

Idealized description

Always on or above and horizontal axis

The area underneath is exactly 1

Mean and standard deviation

Mean & Median of Density Curve

Median: "equal-areas point"

Mean:"banlance point"

For different types of curves

Symmeric density curve: Mean = Median

Skewed righta: Mean is on the right side of the median

Skewed lift: Mean is on the left side of the median

Normal Distributions

Normal Distribution

These density curves are symmetric, single-peaked and bell-shaped

Density function

Importance

Often a good descriptor for some distribution of real data

Good approximations to results of many chance outcomes

Many statistical inference procedures based on Normal distribution work well for other roughly symmetrical distributions

Empirical Rule

About 68% of the values lie within 1 standard deviation of the mean

μ ± σ.

About 95% of the values lie within 2 standard deviations of the mean

μ ± 2σ.

Almost all (99.7%) of the values lie within 3 standard deviations of the mean

μ ± 3σ.

Standard Normal Distribution

A Normal distribution with mean 0 and standard deviation 1

For any Normal distribution we can perform a linear transformation on the variable to obtain an standard Normal distribution

Standard Normal Calculation

Proportion of observations lie in some range of values

Standard normal table

Assesing Normallity

How well the data fit the empirical rule

Histograms

Stem plots

Box plots

A normal probbility plot

Subtopic

If the points on a Normal probility plot lie close to a straight line, the plot indicates the data are Normal

Outliers appear as points that are far away from the overall pattern of the plot

Systematis deviations from a straight line indicate a non-Normal distribution

Examining Relationships

Response Variable and Explanatory Variable

A response variable measures an outcome of a study

An explanatory variable helps explain or influences changes in a response variable

To identify a response and explanatory variable by specifulying values of one variables in order to see how it affects another variable

Explanatory variable does not necessary causes the change in the response variable

Scatterplots and correlation

The relationship between 2 quantitative varibles measured on the same individuals

Plot the explanaory variable as the horizontal axis, and the response variable on the vertical axis

Interpretion a Scatterplot

Look for overall pattern and for striking deviations from that pattern

Outlier: an individual value that falls outside the overall pattern of the relationship

Describe the pattern by the direction, form and strenght of the relationship

Positive & Negative Association

Positive associated

The above-average values and below-average value tend to occur together

Negative associate

above-average values of one tend to accompany below-average values of the other and vice versa

Categorical Variables in Scatterplots

Subtodisplay the different categories

different plot color

different symbol

Correlation

Measrue dircetion and strength of a linear relationship

Between -1 to 1

r near to 0

weak relationship

r near to 1 or -1

strong linear relationship

Does not measure curved relationship between variables

Least Squares Regression Line

Stright line describes how a response varible y changes sa an explanatory varible x changes

predict the value of y

y = a + bx

Least-Squares Regression

Residuals & Residuals Plot

residual

observed y - predicted y

residual plot

how well the regression line fits the data

should show no obvious pattern for liner relationship

Std of residual

Coefficient of Determination

r^2

% of observations lie on least squares regression line

Outliers and Influential Obserbation in Regression

Lurking Variable

not among the explanatory or response variable in a study

Correlation and Regression Wisdom

Correlation: Measuring Linear Association

Correlations Based on Averaged Data

More about Relationships between Two Variable

Transforming to Achieve Linearity

Exponential Growth Model

Dependent variable = log(y) log(y) = b0 + b1x ŷ = 10b0 + b1x

Power Law Model

Dependent variable = log(y) Independent variable = log(x) log(y)= b0 + b1log(x) ŷ = 10b0 + b1log(x)

Relationships between Categorical Variables

Simpon's Paradox

Establishing Causation

Explaing Association

Explaning Causation

Desinging Samples

Observational Study & Experiment

Ovservational study

observe individuals and measure variables of interest

do not attempt to influence the responses

the effect of one variable on another often fail

explanatory variable is confounded with lurking variable

cheaper

experiment

deliberately impose some treatment on individual

observal the responses

allow to pin down the effects of specific variables of interest

Designing Sample

Population and Sample

population

entire group of individuals

census

census attempts to contact every individual in the entire population

advantage

able to find all characteric of the population accurately

disadvantages

expensive

time causing

sample

part of population that we actually examine in order to gather information

sampling

sampling involves studing a part in order to gain information about the whole

advantages

cheaper

less time needed

may miss out centain characteristics of the population

Voluntary response sampling

people who choose themselves by responding to a general appeal

voluntary response bias

people who feel most strongly about an issue are most likely to respond

Convenience Sampling

individuals who are easiest to reach

undercoverage bias

Probability Sample

random sample

each member of the population is equally likely to be included

Simple Random Sampling

A sample of a given size

every possibel sample is equally likely to be chosen

4 steps to choose SRS

1 label

2 table

3 stopping rule

4 identify sample

systematic sample

first member of the sample is chosen according to the some random procedure

the rest are chosen according to some well-defined pattern

Stratified Random Sampling

subgroups of sample, strata appear in approximately the same proportion in the sample as they do in the population

Cluster Sampling

divide population into groups as clusters

randomly select some of the clusters

all the individuals in the chosen clusters are selected to be in the sample

Multi-Stage Sampling Design

Cautions about Sample Surveys

Sampling Bias

Undercoverage

some part of population being sampled is somehow excluded

voluntary response bias

voluntary response bias

self-selected samples

persons who feel most strongly about an issue are most likely to respond

non-response bias

the possible biases of those who choose not to respond

wording bias

wording of the question influences the response in a systematic way

response bias

not give truthful responses to a question

respondent may fail to understand the question

respondent desires to please the interviewer

the ordering of question may influence the response

Desinging Experiments

Design of Experiment

Do something to individuals in order to observe the response

Control

overall effort to mnimize variability in the way experimental unit are obtained and treated

a group reveives the treatment

another group does not reveive any treatment (control group)

compares the responses in reatment group and control group

reduce the problems posed by confounding and lurking variables

placebo

a dummy treatment

Replication

natural variability among the experiment units

reduce the role of variation

increase the sensitivity of the experiment to differences between treatments

Randomizations

divide experimental units into groups by SRS

randomized comparative experiment

ensure that influences other than the treatments operate equally on all groups

completely randomized design

all the experimental units are allocated at random among all treatments

statistically significant

ovserved effect is very large

would rarely occur by chance

Block Design

a group of experimental untis or subjects similar in some way

expected to systematically affect the response to the treatments

random assignment of units to treatments is carried out separately within a block

characteristics

formed based on teh most important unavoidable sources of variability among the experimental units

another form of control, which controls the effects of some outside variables by bring those variables into the experiment to form the blocks

can have any size

chosen based on the likelihood

allows to draw separate conclusions about each block

Matched Pairs Design

an example of block design

compare tow treatments and the subjects are matched in pairs

more similar than unmatched subjects => more effective

Cautions abut Experiment

double-blind experiment

lacking of realism limits ability to apply the conclusions of an experiment to the settings of greater interest