Business Stats

Data is described in 3 ways:

Tables: (freq distribution, Graphs (Bar charts, Histogram), Numerical Measures (Measures of central tendency; location, i.e perecntiles; dispersion)

Position of median

Mean of Sample Data

X bar

Mean of population Data

Meu

what do percentiles do?

Give the relative position/location of the element in the dataset

Percentile Location formula

i = percentile location, P = percentile

Element location for a percentile formula (when i isn’t an integer)

first element, found using location from i, + (difference between 2nd element - first element)* (the decimal component of i)

Q1

25th percentile

Q3

75th percentile

IQR

Q3-Q1

Upper Limit of boxplot

Q3+1.5*IQR

Lower limit of boxplot

Q1- 1.5*IQR

Any data beyond the limits are:

Outliers

Measure of Dispersion + Ex of it

Tell us the spread of the data (do data values remain consistent or do they change a lot) Ex: STD DEV

If range is low:

Values are relatively close and there’s less fluctuation in data

STD Dev (def + formula)

Tells us how far away from the mean data values are + square root of variance

STD Dev of sample formula

sqr root of:

STD Dev of population formula

Sqr Root of:

The higher the std dev:

further away values are from the mean

random experiment

Outcome of which can’t be predicted with certainty

Random variable (denoted by?)

All possible values based on the random experiment; X

Probability

Discrete Random variables

Values obtained by Counting (ex: # of siblings)

Continous Random Variable

Values obtained by measuring (ex: height)

Characteristics of Discrete Probability Distribution

• RV will be discrete

• Prob value is given by formula

Conditions for Binomial Prob Distribution

• RV will be discrete

• IITF

IITF

Identical trials, Independent Trials, Two outcomes for each trial (success or failure), Fixed probabilities for both out outcomes in every trial

Binomial Formula

Types of Continous Probability Distributions

Uniform, Normal

Properties for a graph of a continous probability distribution

Shape is the same as a histogram, Probability = Area under graph, total area under a graph = 1

Properties of a graph of a uniform distribution

Shape is rectangular, probability = area under graph = 1

What’s the probability that flight time is exactly 136.111 mins?

Why is P(120 < x < 130) = P (120≤ x ≤ 130) true?

Bc the lines P(X=120) and P(X=130) are 0

Properties of Normal Distribution

Bell shape, Probability = Area = 1, Mean = Median, Empirical Rule

Empirical Rule

68% of area within 1 STD DEV, 95% of area within 2 STD DEVs, 99% of area within 3 STD DEVs

Z-Score formula

Z-Score values show the area of the Z-Score to the…?

Left

Inferential Statistics

Understanding a population using a sample

Random Sampling

Every population unit is given an equal chance to be part of the sample

Point Estimator + examples

Estimate is a specific point

Mean of Sample Means = ?

Mean of Population (i.e Meu)

Std Dev of Sample Means = ?

Shape of the distribution of a sample mean (x-bar) is normal when:

• n (i.e. sample size) is greater or equal to 30

• Population is normally distributed

For finding probabilities of a normal distribution of sample means, the z-score formula is:

Proportion

(number of observations with a characteristic) / (total number of observations)

P-hat

sample proportion

P

Population Proportion

Mean of Sample Proportions = __

Population Proportion (i.e: P)

STD DEV of Sample Proportion

Note that P in this formula is Population Proportion

The shape of the distribution of sample proportions is normal when?

• np > 5

AND

• n(1-p) >5

n= sample size

p= population proportion

The z-score formula for normally distributed sample proportions =

What’s ∝ (alpha)

The significance level (i.e the chance of being wrong)

What’s a confidence level

1-∝ (How confident we are that we will be correct)

Formula for confidence interval

What’s a critical value?

the z-value used for calculating confidence intervals

Margin of Error formula

Similarites of T-distributions and Z-distributions

Both have a symmetrical “bell” shape, mean = 0, CLT applies for both, area under curve is 1

Differences between T-Distributions and Z-Distributions

For a t-distribution: the probability value depends on the sample size (i.e n-1, aka the degrees of freedom) and the empirical rule doesn’t apply

Formula for confidence interval when pop std dev isn’t known

To find critical value use T-dist

T-dist show area to the _____?

right

Interval estimate formula for a proportion:

Use z-dist for critical value

What are the aspects of a null hypothesis?

Null hypothesis must be either: ≤,≥, or =

What are the aspects of an alternative hypothesis?

Alternative hypothsis must match the sign of the null hypothesis with (<,>, or =)

Hypothesis are about ______ and not about ______?

Population parameters; sample statistics

Between the null and alternative hypothesis, only _____ can be true?

one

Two-tailed test

H_a: U ≠ 23

Lower Tail

H_a: U < 23

Upper tail

H_a: U > 23

Steps for hypothesis testing:

For hypothesis testing when σ (pop. std dev) isn’t known what do we do?

Use a t-dist and use S (sample std dev)

For hypothesis testing what does ∝ correspond to?

Rejection area

CV Method for hypothesis testing:

Find the critical value for the rejection area(s), see whether the test statistic from step 4 is in the non-rejection or rejection area of the curve

P-value method for hypothesis testing:

Find the probability value for the test statistic (direction of p-value based on H_a’s sign (i.e <,>,≠)), see whether said area is greater than or less than ∝, if less than ∝, reject H_o

The mean/average of the difference between sample means (x̄₁, x̄₂ ) is equal to?

μ₁ - μ₂ (difference between population means)

Std dev of (x̄₁ - x̄₂ )

CLT for (x̄₁ - x̄₂) | Dist for (x̄₁ - x̄₂) will be normal if:

• both n₁ and n₂ ≥ 30

OR

• both populations are normal

Z- Distribution of (x̄₁ - x̄₂)

Z formula for (x̄₁ - x̄₂)

σ₁ = Std of population 1

σ₂= Std of population 2

When std devs of population 1 and population aren’t known what formula do we use?

T-Test formula

Formula for DF for Hypothesis testing for 2 populations

What do we always do for DF of 2 Population Hypothesis test?

Round down

What are linear regression used for?

To predict or explain the value of a variable using other variables

A dependent variable is:

always quantitative

Independent variables can be:

quantitative or categorical

Simple linear regression

• one x-variable

• one y-variable

Multiple linear regression

• 2 or more x-variables

• one y-variable

1st condition of linear relationship

There must be a linear relationship between variables for linear regression to work

Equation of a line

If we’re using populations for a linear regression, we use

Beto-0 and Beta-1 instead of b₀ and b₁

SS_xy formula:

SS_x formula:

b₁ formula:

b₀ formula: