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Plan

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Process

Discuss

What sort of place do you live in?

Plan

Collect

Process

Discuss

Where do your friends live?

What sort of place do you live in?

Plan

Collect

Process

Discuss

The Problem Solving Approach

Plan

Discuss

Collect

Process

Plan

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Process

Discuss

Plan

What proportion of students are freshers?

Do most freshers live in university accommodation?

How can you find out?

Who should you ask?

What should you ask them?

Collect

Process

Discuss

Plan

Do third years live in hall?

Do freshers prefer private accommodation?

Do students pay a lot of rent?

Use a questionnaire?

Does the average rent paid depend on type of accommodation?

Collect

Process

Discuss

Plan

- Develop a model of the population.
- One variable may depend on another.
- Turn the model into precise statistical
hypotheses (null and alternative).

H0:

H1:

The average rent is the same for all types of accommodation

The average rents differ

Process

Discuss

Plan

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The questionnaire

Process

Discuss

Plan

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Plan

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Process

Discuss

The Problem Solving Approach

You are

now here.

Plan

Discuss

Collect

Process

Plan

Process

Discuss

Collect

Which data

You did this in your first seminars

Students at three other UK universities have completed the questionnaire

Plan

Collect

Process

Discuss

The Problem Solving Approach

Plan

Discuss

Collect

You are

now here.

Process

Plan

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Discuss

Process

Analysis of variance

- Testing for
- Differences between means

AN

O

VA

ANOVA

If we want to detect differences between means

why do we look at variances?

Plan

Collect

Discuss

Assume the variance σ2 is the same for all

Process

H0 true

The null hypothesis is

H0 : the means are all equal

or μ1 = μ2 = μ3 …… = μk

The alternative hypothesis is

H1 : the means are not all equal

or μi ≠ μj for some i and j

H1 true

Assume H0 true when H1 true

Plan

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Discuss

ANOVA

Process

H0 : μ1 = μ2 = μ3 …… = μk

Average the variances from these

H1 : μi ≠ μj for some i and j

And compare it with the variance from this

Plan

Collect

Discuss

ANOVA

Process

H0 : μ1 = μ2 = μ3 …… = μk

Average the variances from these

H1 : μi ≠ μj for some i and j

Within groups

And compare it with the variance from this

Between groups

A big difference would support H1

Plan

Collect

Discuss

ANOVA

Process

- Measuring the variability within groups
- Calculate individual sample means
- Calculate sum of squared deviations from these sample means for each group

- Calculate the total sum of squares across all groups the within groups sum of squares

4. Then divide by the degrees of freedom n-k

The within group mean square

Plan

Collect

Discuss

ANOVA

Process

- Measuring the variability between groups
- Calculate individual sample means
- Calculate the overall sample mean

To estimate we would use

because

So to estimate σ2 we use

- Calculate the between groups sum of squares as

4. Then divide by the degrees of freedom k-1

The between groups mean square

Plan

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Discuss

The test statistic

Process

We have two estimates of σ2

The within group mean square

The between groups mean square

If H0 is true they should be similar

The test statistic

We look at

k-1 numerator

AND

F has an F distribution with

n-k denominator

degrees of freedom

Plan

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Discuss

Different F distributions

Process

3 and 179 d.f.

3 and 10 d.f.

10 and 10 d.f.

100 and 179 d.f.

Plan

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Discuss

Process

Discussion

H0 : μ1 = μ2 = μ3

H1 : μi ≠ μj for some i and j

α = 0.05

2 and 232 df

The decision rule is Reject if

Do not reject

Can obtain this from Minitab

Reject H0

Plan

Collect

Discuss

Process

Which processes

A random sample of n=235 students across 3 universities was taken

Type of accommodation classified as home, university or private

ANOVA table forrent versus type of accommodation

Numerator df = k - 1 = 2

Denominator df = n - k = 232

Plan

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Process

Which processes

From Minitab

Rent versus type

between groups mean square MSB

within group mean square MSW

Plan

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Discuss

The Problem Solving Approach

Plan

Discuss

Collect

Process

You are

now here.

Plan

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Process

Discuss

DISCUSS

Discussion

H0 : μ1 = μ2 = μ3

H1 : μi ≠ μj for some i and j

α = 0.05

2 and 232 df

The decision rule is Reject if

From sample data

Do not reject

Reject H0

Plan

Collect

Process

Discuss

Can we say why?

Individual 95% CIs For Mean Based on Pooled StDev

Level N Mean StDev ----+---------+---------+---------+-----

Home 5 319.60 251.78 (----------------*----------------)

Private 117 248.36 85.78 (---*--)

Uni 113 292.82 94.34 (---*--)

----+---------+---------+---------+-----

250 300 350 400

Pooled StDev = 95.23

What about students living at home?

Have we asked the right questions?

Plan

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Process

Discuss

What can we conclude?

- Other questions?
- about what is this due to
- about other questions/relationships

Plan

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Process

Discuss

The Problem Solving Approach

You can build on the first try by continuing here...

Plan

Discuss

Collect

You are

now here.

Process

Have you got all the evidence you want?