Griffiths D. Head First Statistics

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the χ

distribution

Step 3: Determine the critical region for your decision.

You’ll need to use the significance level and number of degrees of freedom

Step 4: Calculate the test statistic Χ

You can calculate this using your observed and expected frequencies from step 2.

Step 5: See whether the test statistic is within the critical region.

Step 6: Make your decision.

582 Chapter 14

Fat Dan thinks that the dice in the dice games are loaded. Take a look at the following observed

frequencies for one six-sided die, and test whether there’s enough evidence to support the claim

that the die isn’t fair at the 1% significance level. We’ll guide you through the steps.

Here are the observed frequencies:

x Observed frequency Expected frequency

107

167

198

167

192

167

125

167

132

167

248

167

Value 1 2 3 4 5 6

Frequency

107 198 192 125 132 248

Step 1: Decide on the hypothesis you’re going to test, and its alternative.

To test whether the die is fair, we have to determine whether there’s sufficient evidence that it isn’t.

This gives you

: The die is fair, and every value has an equal chance of being thrown. This means the probability of

getting each value is 1/6.

: The die isn’t fair.

Step 2: Find the expected frequencies and the degrees of freedom.

Start off by completing the expected frequencies for the die. You’ll need to take into account how many times the die is

thrown in total, and the probability of getting each value. X represents the value of one toss of the die.

The total expected frequency

needs to match the total

observed frequency. If you add

the observed frequencies together,

you get 1002.

The probability of getting each

value is 1/6. This means the

expected frequency of each value

is 1002/6 = 167.

Once you’ve found the expected frequencies, what are the number of degrees of freedom?

We had to find 6 expected frequencies, and their total had to equal 1002. In other words, we had to find 6

pieces of information with 1 restriction. This gives us

ν = 6 - 1

= 5

long exercise solution

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the χ

distribution

Step 3: Determine the critical region for your decision.

You’ll need to use the significance level and number of degrees of freedom

The critical region is given by X

> 15.09. As X

= 88.24, the test statistic is within the critical region.

From probability tables, χ

(5) = 15.09. This means that the critical region is where X

> 15.09.

Step 4: Calculate the test statistic Χ

You can calculate this using your observed and expected frequencies from step 2.

(O - E)

= (107-167)

/167 + (198-167)

/167 + (192-167)

/167 + (125-167)

/167 + (132-167)

/167 + (248-167)

/167

= (-60)

/167 + (31)

/167 + (25)

/167 + (-42)

/167 + (-35)

/167 + (81)

/167

= (3600 + 961 + 625 + 1764 + 1225 + 6561)/167

= 14736/167

= 88.24

Step 5: See whether the test statistic is within the critical region.

As your test statistic lies within the critical region, this means that there is sufficient evidence at the 1% level

to reject the null hypothesis. In other words, you accept the alternate hypothesis that the die isn’t fair.

Step 6: Make your decision.

584 Chapter 14

So can you use the χ

goodness

of fit test with any underlying

probability distribution?

The χ

goodness of fit test works for pretty much any

probability distribution.

You can use the χ

distribution to test the goodness of fit of any probability

distribution, just as long as you have a set of observed frequencies, and you can

work out what you expect the frequencies to be.

The hardest thing is working out what the degrees of freedom for ν should be.

Here are the degrees of freedom for some of the most common probability

distributions you’ll want to use with the χ

goodness of fit.

Distribution Condition

Binomial

You know what p is

You don’t know what p is, and

you have to estimate it from

the observed frequencies

ν = n - 1

ν = n - 2

Poisson

You know what λ is

You don’t know what λ is, and

you have to estimate it from

the observed frequencies

ν = n - 1

ν = n - 2

Normal

You know what μ and σ

are

You don’t know what μ and σ

are, and you have to estimate

them from the observed

frequencies

ν = n - 1

ν = n - 3

p is the probability of success, or

the proportion of successes in the

population.

λ is the rate of

occurrences in an

interval.

n is the

total number

of observed

frequencies.

distribution cheat sheet

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the χ

distribution

Fat Dan has another problem

So far you’ve investigated whether the slot machines seem to be rigged

in some way, by using a goodness of fit test to see whether the observed

frequencies you have correspond to the expected probability distribution.

Fat Dan has other problems, though, and this time it’s his staff.

Fat Dan thinks he’s losing more money than he should from one of the

croupiers on the blackjack tables. Can you determine whether there’s

significant evidence to show whether or not Fat Dan’s right?

Here are the three croupiers who man the tables:

Croupier A Croupier B Croupier C

Win

43 49 22

Draw

8 2 5

Lose

47 44 30

What we need is some way of testing whether the outcome of the game

is dependent on which croupier is leading the game.

What do you need to know in order to test this hypothesis?

These are the

outcomes you get for

each of the games.

These are the observed

results for each of

the croupiers.

586 Chapter 14

So far we’ve looked at the χ

distribution in terms of performing goodness

of fit tests. This isn’t the only use of the χ

distribution. The χ

distribution

can also be used to perform tests of independence.

A χ

test for independence is a test to see whether two factors are

independent, or whether there seems to be some sort of association

between them. This is just the situation we have with the croupiers. We

want to test whether the croupier leading a game of blackjack has any

impact on the outcome. In other words, we assume that the choice of

croupier is independent of the outcome, unless there’s sufficient evidence

against it.

You conduct a test for independence in the same way you conduct a

goodness of fit test. You set up a hypothesis, use the observed and expected

frequencies to calculate the Χ

test statistic, and then see if it falls within the

critical region.

Now hold it right there! I think

you’re missing something. How can we

work out the expected frequencies?

All we have to go on is the observed

frequencies the actual game outcomes.

We need to know what the expected frequencies are

in order to calculate the test statistic Χ

This means that we need some way of calculating the expected frequencies

from the observed frequencies. And it all comes down to probability...

the χ

distribution can test for independence

testing for independence

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the χ

distribution

There are a few steps you need to go through to find the expected

frequencies.

To start off, calculate the total frequencies for the outcomes and the croupiers,

and also the grand total. You can show the results in a table like this, called a

contingency table.

Croupier A Croupier B Croupier C Total

Win

43 49 22 114

Draw

8 2 5 15

Lose

47 44 30 121

Total

98 95 57 250

Now we can use this information to find the expected number of wins for each croupier.

Let’s start by finding the expected frequency for the number of wins with croupier A.

First off, we can use these grand totals to find the probability of getting a particular

outcome, or a particular croupier. As an example, to find the probability of winning, you

divide the total number of wins by the grand total:

P(Win) = Total Wins

Grand T

otal

P(A) = Total A

Grand Total

Now if the croupier and the outcome of the game are independent, as we

assume they are, this means that you can find the probability of getting

a win with croupier A by multiplying together these two probabilities. In

other words:

P(Win and A) = Total Wins

Total A

and Total Grand Total

How can we use this to find the expected number of wins for croupier A?

Similarly, you can find the probability of playing against croupier A by

dividing the total for croupier A by the grand total.

Back in chapter 4, we saw

that for independent events,

P(A ∩ B) = P(A) x P(B).

Total for

croupier A

Total number

of wins

Grand total

You can find the expected frequencies using probability

588 Chapter 14

(O - E)

So what are the frequencies?

So far, we’ve found that the probability of winning with croupier A, and

we want to use this to find the expected frequency of wins. To do this, all

we just need to multiply the probability of winning with croupier A by the

grand total. This gives us

Expected frequency = Row Total

Column Total

Grand Total

Expected frequency = Grand Total

Total Wins

Total A

Grand Total Grand Total

= Total Wins

Total A

Grand Total

In other words, to find the expected frequency of wins with croupier A,

multiply the total number of wins by the total number of games with

croupier A, and divide by the grand total.

How do we find the frequencies in general?

You can generalize this so that you have a nice, easy result you can apply

to every frequency you need to find. To find the expected frequency for a

particular row and column combination, multiply the total for the row by

the total for the column, and divide by the grand total.

Once you’ve figured out what all the expected frequencies are, you can use

this to calculate the test statistic Χ

. It’s the same test statistic as before, so

you need to calculate

The key is to ensure you include every observed frequency and every

corresponding expected frequency.

For every observed frequency, subtract

the expected frequency, square the result,

and divide by the expected frequency.

Then add your results together.

finding the frequencies

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the χ

distribution

Here’s the table showing the observed frequencies for the croupiers. Your task is to figure out all

the expected frequencies.

Once you’ve found all the expected frequencies, calculate the test statistic Χ

. Use the table below to help you. The

first column gives all the observed frequencies, the second column is for the corresponding expected frequencies,

and if you add together all the numbers in the third column, it gives you your test statistic.

Croupier A Croupier B Croupier C Total

Win

43 49 22 114

Draw

8 2 5 15

Lose

47 44 30 121

Total

98 95 57 250

Croupier A Croupier B Croupier C

Win

(114

98)/250=44.688

Draw

(15

98)/250=5.88

Lose

(121

98)/250=47.432

Observed Expected (O - E)

43 44.688 (43-44.688)

/44.688 = 2.85/44.688 = 0.064

8 5.88 (8-5.88)

/5.88 = 4.4944/5.88 = 0.764

47 47.432 (47-47.432)

/47.432 = 0.187/47.432 = 0.004

ΣO = 250

ΣE =

These are

the observed

frequencies.

Work out

each of the

expected

frequencies

here.

(Row total x column total)/grand total

Use the values in the first two

columns to help you calculate this.

(O - E)

590 Chapter 14

Here’s the table showing the observed frequencies for the croupiers. Your task is to figure out all

the expected frequencies.

Once you’ve found all the expected frequencies, calculate the test statistic Χ

. Use the table below to help you. The

first column gives all the observed frequencies, the second column is for the corresponding expected frequencies,

and if you add together all the numbers in the third column, it gives you your test statistic.

Croupier A Croupier B Croupier C Total

Win

43 49 22 114

Draw

8 2 5 15

Lose

47 44 30 121

Total

98 95 57 250

Croupier A Croupier B Croupier C

Win

(114

98)/250=44.688

(114x95)/250=43.32 (114x57)/250=25.992

Draw

(15

98)/250=5.88

(15x95)/250=5.7 (15x57)/250=3.42

Lose

(121

98)/250=47.432

(121x95)/250=45.98 (121x57)/250=27.588

Observed Expected (O - E)

43 44.688 (43-44.688)

/44.688 = 2.85/44.688 = 0.064

8 5.88 (8-5.88)

/5.88 = 4.4944/5.88 = 0.764

47 47.432 (47-47.432)

/47.432 = 0.187/47.432 = 0.004

43.32 (49-43.32)

/43.32 = 5.68/43.32 = 0.131

5.7 (2-5.7)

/5.7 = 13.69/5.7 = 2.402

45.98 (44-45.98)

/45.98 = 3.9204/45.98 = 0.085

25.992 (22-25.992)

/25.992 = 15.936/25.992 = 0.613

3.42 (5-3.42)

/3.42 = 2.4964/3.42 = 0.730

27.588 (30-27.588)

/27.588 = 5.817/27.588 = 0.211

ΣO = 250

ΣE =

250

Observed

frequencies

Expected

frequencies

This is your test statistic

exercise solution

(O - E)

5.004