# Statistics 2

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• Created by: xooley
• Created on: 12-05-16 18:49

## Chapter 1 - Discrete probability distributions

Mean = E(X) = Σ(xP(X=x))

Variance = Var(X) = E(X²) - E(X)²

Standard deviation = sqrt(Var(X))

E(a) = a

E(aX) = aE(X)

E(aX+b) = aE(X)+b

Var(a) = 0

Var(aX) = a² Var(X)

Var(aX+b) = a² Var(X)

if X~B(n,p) then: E(X) = np

Var(X) = np(1-p)

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## Chapter 2 - Poisson distribution

Independent, Random, At a constant average rate, No upper limit - IRAN

For probabilities not in the table and the mean and the variance, look in the front of the formula book (page 11)

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## Chapter 3 - Continuous probability distributions

F(x) = integral f(x)

f(x) = dy/dx F(x)

The nth percentile = F(N) = n/100

Quartiles: lower = 0.25, upper = 0.75

E(X) = integral(xf(x))

E(X²) = integral(x²f(x))

Rectangular distributions formulas are in the formula booklet.

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## Chapter 4 - Confidence intervals

An estimate of a population standard deviance calculated from a random sample of size n has n-1 degrees of freedom.

A confidence interval is calculated with:
x̄±(t or z)(sd/√n)

If the standard deviation is known, we use z tables
If the standard deviation is unknown, we use t tables

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## Chapter 5 - Hypothesis testing

One-tailed test - where the H1 hypothesis is testing for an increase or decrease

Two-tailed test - where the H1 hypothesis is testing for a change with no direction

Test statistic = (x̅-μ) / (σ/√n)

Type 1 error - where H0 is rejected but it is true

Type 2 error - where H0 is accepted but it is false

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## Chapter 6 - Further hypothesis testing for means

To carry out a hypothesis test for a mean based on a large sample from an unspecified distribution: the test statistic = (x̅-μ) / (σ/√n) and it is compared to critical z values

To carry out a hypothesis test for a mean based on a sample from a normal distribution with an unknown standard deviation: the test statistic = (x̅-μ) / (s/√n) and it is compared to critical t values.

For a sample size of n, v = n-1

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## Chapter 7 - Contingency tables

Expected number = row total x column total / grand total

X² = Σ(O-E)²/E
provided that the Os are frequencies and the Es are greater than 5

An (mxn) contingency table has (m-1)(n-1) degrees of freedom

If it is necessary to combine two classes to increase the size of the Es the most similar classes should be combined

For a 2x2 table: Σ(|O-E|-0.5)²/E

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