Binomial Distribution - Chapter 1

Probability of an outcome = P

Number of trials of an event = n

Number of occurrences of an outcome = r

where

Use the button on your calculator (pronounced n choose r)

Assume independence X~B(n,p)

Sampling – Chapter 4

a) A sample of parent population must be random

b) A sample must be large enough to be representative

c) Simple random sample – every population member has an equal chance of being chosen

d) A population made of sub groups (e.g. sex, age) needs a stratified sample. The number chosen from each stratum is proportional to the size of the stratum.

Continuous Random Variables – Chapter 2

See sheet previously provided

Expectation and Variance – Chapter 7

Expected average

Variance = Var(x) or V(x)

Normal Distribution – Chapter 3

X~N(μ,σ²)

Total area = 1

z is the standardised value of observation X

Φ(z) = P(z ≤ Z)

P(z ≤ 1.3) = Φ(1.3)

P(z ≤ 1.3) = 1-Φ(1.3)

P(z ≤ -1.3) = 1-Φ(1.3)

P(z ≤ -1.3) = 1-[1-Φ(1.3)] = Φ(1.3)

Normal → Binomial (approximation) Chapter 5

When p=½ and any n or for large values of n (n>25).

X~B(n,p) → Y~N(np, np(1-p)) where np=μ and np(1-p)=σ²

Discrete → Continuous

●Remember.

Discrete (Binomial) Continuous (Normal)

P(x > 50) → P(Y > 50.5)

P(x < 50) → P(Y > 49.5)

P(30 < x < 60) → P(30.5 < Y < 59.5)

P(30 ≤ x ≤ 60) → P(29.5 < Y < 60.5)

Sampling distribution of mean - Chapter 6

sample mean

Distribution of sample mean is like a normal distribution. For sample size n, mean is μ, variance is σ²/n i.e.

Standard deviation of i.e. is standard error.

Central limit theorem

● the distribution of sample means are approx. normal if sample size is > 30

● the variance is

Unbiased estimates of μ & σ² - Chapter 6

When we don’t know μ & σ² of parent population, we make estimates using our sample.

- sample mean (unbiased)

is an unbiased estimator of σ² (s² is sample variance)

Confidence - Chapter 6

The likelihood that the population mean (μ) lies within a given interval.

90% μ ± 1.645 s.e.

95% μ ± 1.96 s.e.

99% μ ± 2.575 s.e.

68% μ ± 1 s.e.

Remember parent population σ is unknown so simply multiply by