2 -tests

2The significance test2There are two main situations where a significance test is used:

a is used when you have some practical data and you

want to know how well a particular statistical distri

2 goodness- of - fit test

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bution, e.g. binomial or normal,

models the data. The null hypothesis is that the particular distribution does provide

a model for the data; the alternative hypothesis is that it does not.

a

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H

(or ) is used when you have some practical

data concerning two variables and you want to know whether they are independent

or whether there is an association bet

2 test for independence association

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ween them. The null hypothesis is that the

factors are independent; the alternative hypothesis is that they are not.

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H

The null hypothesis is assumed true and we calculate expected frequencies,

denoted or , based on this assumption. The expected frequencies are compared

with the actual (or observed) frequencies, deneE f

oted or .oO f

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2

A test statistic involving and is calculated. This is usually denoted

and, subject to certain conditions, it can be approximated by a distribution.

O E X

2The distribution2The distribution has one parameter, , and the shape of the distribution is different

for different values of .

2

12 2

1

02

( ) , d

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The p.d.f for the distribution with parameter is

x

z tx eP x z t e t

2

1 2

2

Features of the distribution:

It is reverse -shaped for and .

It is positively-skewed for .

The larger the value of , the more symmetric the distribution becomes.

When is large, t

J

he distribution is approximately normal.

The parameter is known as the number of degrees of freedom and it is the number

of independent variables used in calculating the test statistic. Finding depends on

the particular test being applied.

2The table2

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The test is conducted as a one-tailed (upper tail) test.

If the test value lies in the critical region, the the null hypothesis

is rejected in favour of the alternative hypothesis .

The critical

H

H

25%

5%

value depends on the level of significance of the test.

Note that for a level of significance the critical value may be written

as for a particular value of .

2Performing a goodness-of-fit test

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For a set of data with observed frequencies :

Make the null hypothesis that the data are distributed in a particular way and the

alternative hypothesis that they are not.

Calculate ,

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H

E

1.

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the frequencies expected if the distribution follows the one given in .

If for any class, combine adjacent classes to form a class that is sufficiently

large.

Work out the number of degr

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3.

2

2

2

ees of freedom, , where

number of classes number of restrictions

Decide on the level of the test and look up the appropriate critical value in the table.

Calculate .

Comp

O EX

E

4.

5. 20 0are with the critical value. Make your conclusion ( is rejected or is not

rejected) and relate it to the context of the question.

X H H

Particular attention should be given to the combining of classes in Step 2

and the calculation of in Step 3.Note :

Goodness-of-fit tests for particular distributions

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Uniform distribution

n Test 1:

Read Example 12.1, pp.567-568

Do Q1, Q2, Q5, Q10, Q11, Q12, pp.569-570

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Distribution in a given ratio

n Test 2 :

Read Example 12.2, pp.568-569

Do Q3, Q4, Q6, Q9, pp.569-570

Goodness-of-fit tests for particular distributions

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2

Binomial distribution

(a) if is known then

(b) if is unknown and estimated from the observed

frequencies using then

p n

p

x np n

Test 3 :

Read Example 12.3, pp.571-572

Do Q1-Q7, pp.579-580

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2

Poisson distribution

(a) if is known then

(b) if is unknown and estimated from the observed

frequencies using then

n

x n

Test 4 :

Read Example 12.4, 12.5, pp.573-575

Do Q8-Q14, pp.580-581

Goodness-of-fit tests for particular distributions

2

2

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3

Normal distribution

(a) if and are known then

(b) if and are unknown and estimated from the observed

frequencies then

n

n

Test 5 :

Read Example 12.6, 12.7, pp.576-578

Do Q15-Q17, p.582

2The significance test for independenceThis test is used when data are classified according to two different factors or

attributes and these are often displayed in a table, known as a .contingency table

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This data displays examination grades in

three American further education colleges.

This is an example of a contingency

table ( rows and columns).

2A test is used to investigate whether the two factors (here, "examination grade"

and "college") are independent or whether there is an association between them.

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The test follows a similar pattern to the goodness-of-fit test, but now the null

hypothesis is that the two factors are independent and the alternative hypothesis

is that there is an association b

H H

etween them.

Special attention should be given to the calculation of , the expected frequencies,

and , the degrees of freedom.

E

2The significance test for independence

row total column totalexpected frequency

grand totalE

5 cells should be combined if .E Note :

1 1For a contingency table, h k h k

For CIE A-Level Further Mathematics Yate's correction is

required.

Note :

not

Read Example 12.8, pp.583-585; Example 12.11, pp.592-593

Do Exercise 12C, pp.588-590