Aims:• To be able to decide which distribution to use when constructing a C.I. • To practice exam style questions.• To be able to calculate the unbiased estimates for the mean and variance.

Estimation Lesson 3

ParentDistribution

Variance Assumptions or Theory for

C I

NormalX~(µ,σ2)

Known

Not normal Known

Not normal

NormalX~(µ,σ2)

n

zxn

zx ,

nszx

nszx ,

1 1,n ns sx t x tn n

2__2

2

1x

nx

nns

X

n

zxn

zx ,

Unknown

2__2

2

1x

nx

nns

Unknown

Summary table of which CI formula to use

2~ ,

no CLT

n

X N n

2

30

~ ,

CLT yes

n

X N n

2

30

~ ,

CLT yes

n

sX N n

2

30

~ ,

CLT no

n

sX T n

Note: If sample is < 30 and not from a normal distribution we can’t use either the z or t distribution to model the situation

Unbiased Estimates of the Mean & Variance

The variance if unknown can be calculated by using the formula below and we call it s2 rather than σ2

2

222 1 s1 1

xn x x xn n n

The unbiased estimate of the mean from a sample is calculated using:

nx

x

Jan 08 Exam Questionof Nadal’s

of Nadal’s

on Nadal’s

6 mins have a go!

1. Complete relay race2. Do revision Exercise page

107. Qu 1 to 5 compulsory, 5 to 9 optional