MGSC 291 Exam 1 (Walters)

Descriptive Statistics

Collecting, summarizing, and presenting data

Inferential Statistics

drawing conclusions about a population based on sample data from that population

Parameter

a number used to describe a population

Statistic

a number calculated from a sample and is used to estimate the parameter

Time Series

variables that are measured at regular intervals over time (hourly, daily, weekly, monthly, quarterly, annual...)

Cross-Sectional Data

several variables are all measured at the same time point (or time frame)

Big Data

Data warehouses that have grown enormously in size, with the use of powerful computers, the information contained in them is accessible and used to help make decisions

Discrete variables

jumps between possible values

continuous variable`

another possible value between any two values

Nominal variables

categorical variables for which the categories do not have a natural ordering

Ordinal Variables

categorical variables for which the categories have a natural ordering

boxplots

For quantitative data

works for small to large datasets

plots the five number summary

great for side-by-side comparisons

R code for boxplots

boxplot()

Histogram

medium to large quantitative datasets

Bins touch

Choice of number of bins can distort features of the shape of the distribution

R code for Histograms

hist()

Line graph

Displays quantitative data changing over time

Time on x axis (horizontal)

Variable on y axis (vertical)

Bar Graph

Use for qualitative data

Can be horizontal or vertical

can display parts of a whole or separate values

For nominal data: put in ascending/descending order

For ordinal data: put in order of categories

Pie Charts

Categorical Data

Not Good for comparisons

Do not use

Scatterplot

Used to depict two potentially related variables

Each point is a pairing

Linear, curvilinear, or no relationship

Positive vs negative relationship

Sample Mean

Average of sample

Median

50th percentile of the sample

Middle observation in the ordered list if "n" is odd

Average of 2 middle observations if "n" is even

If data is skewed to the right:

mean > median

If data is skewed to the left:

mean

Coefficient of Variation

Standard deviation expressed as a percent of the mean (( sd/mean) x 100)

Chebyshev's Inequality

1-1/k^2

Empirical Rule

The rules gives the approximate % of observations w/in 1 standard deviation (68%), 2 standard deviations (95%) and 3 standard deviations (99.7%) of the mean when the histogram is well approx. by a normal curve

Z-Score

(x-mean)/SD

Kurtosis

"tailedness" of the distribution

Complement Rule

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

Union Rule

P(A U B) = P(A) + P(B) - P(A ∩ B)

Two events are disjoint (mutually exclusive) if ____

they share no outcome in common (P(A ∩ B) = 0)

Checking for Independence

Is P(A|B) = P(A) ?

Is P(B|A) = P(B) ?

Is P(A and B) = P(A) P(B) ?

Bayes' Theorem

A theorem that enables the use of sample information to revise prior probabilities.

Discrete Random variable (probability mass function and cumulative distribution function)

f(x) = P(X=x)

F(x) = P(X=

Binomial Experiment

A fixed number, "n", of trials:

1. Each trial is independent and indentical of the others.

2. There are only two outcomes.

3. The probability of each outcome remains constant from trial to trial.

R code for binomial probabilities

dbinom(j,n,p) = P(Y=j)

pbinom(J,n,p) = P(Y=0) + P(Y=1) + P(Y+J) or P(Y =< or => J)

Formula for probability distribution for a binomial experiment

(n!/x!(n-x)!) * p^x * (1-p)^n-x

Binomial random variable: mean and variance formulas:

mean = n*p

variance = n*p(1-p)

Poisson Probability Distribution

"X" be the number of times an event occurs during an interval of fixed length "t". "X" is this distribution if:

1. The event cannot occur twice at exactly the same time/place/point

2. No occurrence of the event being analyzed affects that probability of the event re-occurring

3. The expected number of occurrences of the event during any such interval is constant

DESIGNED TO BE APPLIED TO EVENTS THAT OCCUR RELATIVELY RARELY

Formula for Poisson Distribution:

((mean^x * e^-mean) / x!)

R code for Poisson probabilities:

P(X=j) = dpois(j, mean/variance(ʎ))

P(X=

Exponential Distribution

A probability distribution associated with the time between arrivals

Formula(s) for an Exponential Distribution:

mean = 1/ʎ

sd = 1/ʎ

f(x) = ʎe^-ʎx for all x=> 0 and ʎ > 0

F(X) = P(X=

R code for an Exponential Distribution (when x =< t)

pexp(t, ʎ)

R code for Normal distribution and _th percentile

pnorm(x, mean, sd)

qnorm(prob,mean,sd)

R code for Empirical Rule (Find P(y< Z < x))

pnorm(x) - pnorm(y)

observational study

Observes cases and records variables of interest but no treatments are applied by the investigator

experiment study

A study where treatments are applied. The goal is to make a statement regarding cause and effect

Sample Survey

When respondents in a sample provide their own data

Census

When a survey attempts to use the entire population as the sample

Simple Random Sample

A sample of size n selected from the population in such a way that each possible sample of size n has an equal chance of being selected.

stratified random sample

1. Separate the sampling frame into strata

2. Take a separate simple random sample from each strata

Random Cluster Sampling

1. Separate the sampling frame into clusters

2. Randomly sample entire clusters

3. In each selected cluster, either take a census or randomly sample within the clster

Sampling Frame

the list we draw a sample from

Undercoverage

occurs when some groups in the population are left out of the process of choosing the sample

non-response bias

bias introduced to a sample when a large fraction of those sampled fails to respond

Central Limit Theorem (CLT)

The sampling distribution of a sum or percentage will become approximately normal as the sample size gets larger

What has a t-distribution?

(x-mean)/(s/sqrt(n))

*WHEN UNKNOWN σ

What has a normal distribution?

(x-mean)/((σ/sqrt(n))

*WHEN σ IS KNOWN

Confidence interval when σ is known:

sample mean +/- (Z*a/2) * (σ/sqrt(n)

Confidence interval when σ is unknown:

sample mean +/- (t*a/2) * (s/sqrt(n))

R code for Confidence interval:

t.test( x, conf.level)

R code for Confidence interval for p (population proportion)

prop.test(x,y,correct=false)

(EX: 42 out of 118 answer survey to see if OK to use cellphone, find a 95 confidence interval.... x= 42 and y = 118)

subjective probability

is drawn on personal judgment.

empirical probability

when the probability comes from the frequency of the event's occurrence

Classical Probability

based on logical analysis rather than on observation or personal judgment.