Griffiths D. Head First Statistics

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the χ

distribution

Step 4: Calculate the test statistic Χ

. Use the table below to help you.

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

Step 6: Make your decision.

Observed Expected (O - E)

375 382.3 (375-382.3)

/382.3 = 53.29/382.3 = 0.139

379 374.6 (379-374.6)

/374.6 = 19.36/374.6 = 0.005

46 43.1 (46-43.1)

/43.1 = 8.41/43.1 = 0.195

367 353.6 (367-353.6)

/353.6 = 179.56/353.6 = 0.508

336

357

362

ΣO =

ΣE =

(O - E)

602 Chapter 14

Fat Dan thinks that one or more of his croupiers are somehow influencing the results of the roulette

wheel. Here’s data showing the observed frequency with which the ball lands in each color pocket

for each of the croupiers. Conduct a test at the 5% level to see whether pocket color and croupier

are independent, or whether there is sufficient evidence to show there might be something going on.

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

You want to test whether or not pocket color is independent of croupier. This gives

: Roulette wheel pocket color and croupier are independent.

: Pocket color and croupier are not independent.

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

There are 3 columns and 3 rows, and we find the number of degrees of freedom by multiplying together (number

of rows - 1) and (number of columns - 1). This gives us

ν = 2x2

= 4

Step 3: Determine the critical region for your decision.

Croupier A Croupier B Croupier C

Red

375 367 357

Black

379 336 362

Green

46 37 41

Croupier A Croupier B Croupier C Total

Red

1099

800/2300=382.3 1099

740/2300=353.6

1099x760/2300=363.1 1099

Black

1077

800/2300=374.6

1077x740/2300=346.5 1077x760/2300=355.9 1077

Green

124

800/2300=43.1

124x740/2300=39.9 124x760/2300=41.0 124

Total

800

740 760 2300

You find the expected frequencies by multiplying each row and column total, and dividing by the grand total.

From probability tables, χ

(4) = 9.49. This means that the critical region is given by X

> 9.49.

chapter-ending long exercise solution

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the χ

distribution

The critical region is given by X

> 9.48. As X

= 1.583, the test statistic is outside the critical region.

Step 4: Calculate the test statistic Χ

. Use the table below to help you.

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

As your test statistic lies outside the critical region, this means that there is insufficient evidence at the 5%

level to reject the null hypothesis. In other words, you accept the null hypothesis that pocket color and croupier

are independent.

Step 6: Make your decision.

Observed Expected (O - E)

375 382.3 (375-382.3)

/382.3 = 53.29/382.3 = 0.139

379 374.6 (379-374.6)

/374.6 = 19.36/374.6 = 0.005

46 43.1 (46-43.1)

/43.1 = 8.41/43.1 = 0.195

367 353.6 (367-353.6)

/353.6 = 179.56/353.6 = 0.508

336

346.5 (336-346.5)

/346.5 = 110.25/346.5 = 0.318

39.9 (37-39.9)

/39.9 = 8.41/39.9 = 0.211

357

363.1 (357-363.1)

/363.1 = 37.21/363.1 = 0.102

362

355.9 (362-355.9)

/355.9 = 37.21/355.9 = 0.105

41.0 (41-41)

/41 = 0/41 = 0

ΣO = 2300

ΣE = 2300

(O - E)

= 1.583

This means that the test statistic is given by X

= 1.583.

this is a new chapter 605

correlation and regression

What’s My Line?

The more I use this

sandpaper, the less

chance there is of him

noticing my stubble.

Have you ever wondered how two things are connected?

So far we’ve looked at statistics that tell you about just one variable—like men’s height,

points scored by basketball players, or how long gumball flavor lasts—but there are other

statistics that tell you about the connection between variables. Seeing how things are

connected can give you a lot of information about the real world, information that you can

use to your advantage. Stay with us while we show you the key to spotting connections:

correlation and regression.

606 Chapter 15

Never trust the weather

Concerts are best when they’re in the open air—at least that’s

what these groovy guys think. They have a thriving business

organizing open-air concerts, and ticket sales for the summer

look promising.

Today’s concert looks like it will be one of their best ones ever.

The band has just started rehearsing, but there’s a cloud on the

horizon...

Feel that funky

rhythm, baby.

Sweet! But is that

a rain cloud I see

up there?

Before too long the sky’s overcast, temperatures are dipping, and

it looks like rain. Even worse, ticket sales are hit. The guys are in

trouble, and they can’t afford for this to happen again.

What the guys want is to be able to predict what concert attendance

will be given predicted hours of sunshine. That way, they’ll be able to

gauge the impact an overcast day is likely to have on attendance. If

it looks like attendance will fall below 3,500 people, the point where

ticket sales won’t cover expenses, then they’ll cancel the concert

They need your help.

the sunshine-attendance connection

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correlation and regression

That’s easy. We can find the mean

and standard deviation and look at the

distribution. That will tell us everything.

Let’s analyze sunshine and attendance

Here’s sample data showing the predicted hours of sunshine and concert

attendance for different events. How can we use this to estimate ticket

sales based on the predicted hours of sunshine for the day?

Most of the time, that’s exactly the sort of thing we’d

need to do to predict likely outcomes.

The problem this time is, what would we find the mean and standard

deviation of ? Would we use the concert attendance as the basis for our

calculations, or would we use the hours of sunshine? Neither one of them

gives us all the information that we need. Instead of considering just one set of

data, we need to look at both.

So far we’ve looked at independent random variables, but not ones that are

dependent. We can assume that if the weather is poor, the probability of high

attendance at an open air concert will be lower than if the weather is sunny.

But how do we model this connection, and how do we use this to predict

attendance based on hours of sunshine?

It all comes down to the type of data.

How would you go about modelling the connection

between sets of data?

Sunshine (hours)

1.9 2.5 3.2 3.8 4.7 5.5 5.9 7.2

Concert attendance (100’s)

22 33 30 42 38 49 42 55

608 Chapter 15

Exploring types of data

Up until now, the sort of data we’ve been dealing with has been univariate.

Univariate data concerns the frequency or probability of a single variable. As an

example, univariate data could describe the winnings at a casino or the weights of

brides in Statsville. In each case, just one thing is being described.

What univariate data can’t do is show you connections between sets of data. For

example, if you had univariate data describing the attendance figures at an open air

concert, it wouldn’t tell you anything about the predicted hours of sunshine on that

day. It would just give you figures for concert attendance.

So what if we do need to know what the connection is between variables? While

univariate data can’t give us this information, there’s another type of data that

can—bivariate data.

Open Air Concert Attendance

Attendance

Frequency

Univariate data for concert

attendance tells you nothing

about the hours of sunshine.

All about bivariate data

Bivariate data gives you the value of two variables for each observation, not

just one. As an example, it can give you both the predicted hours of sunshine and

the concert attendance for a single event or observation, like this.

Sunshine (hours)

1.9 2.5 3.2 3.8 4.7 5.5 5.9 7.2

Concert attendance (100’s)

22 33 30 42 38 49 42 55

Bivariate data gives you the

value of two variables for each

observation.

If one of the variables has been controlled in some way or is used to explain the

other, it is called the independent or explanatory variable. The other variable

is called the dependent or response variable. In our example, we want to use

sunshine to predict attendance, so sunshine is the independent variable, and

attendance is the dependent.

introducing bivariate data

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correlation and regression

Visualizing bivariate data

Just as with univariate data, you can draw charts for bivariate data to

help you see patterns. Instead of plotting a value against its frequency or

probability, you plot one variable on the x-axis and the other variable against

it on the y-axis. This helps you to visualize the connection between the two

variables.

This sort of chart is called a scatter diagram or scatter plot, and

drawing one of these is a lot like drawing any other sort of chart.

Start off by drawing two axes, one vertical and one horizontal. Use the

x-axis for one variable and the y-axis for the other. The independent variable

normally goes along the x-axis, leaving the dependent variable to go on

the y-axis. Once you’ve drawn your axes, you then take the values for each

observation and plot them on the scatter plot.

Here’s a scatter plot showing the number of hours of sunshine and concert

attendance figures for particular events or observations. As the predicted

number of hours sunshine is the independent variable, we’ve plotted it on

the x-axis. The concert attendance is the dependent variable, so that’s on the

y-axis.

x (sunshine)

1.9 2.5 3.2 3.8 4.7 5.5 5.9 7.2

y (attendance)

22 33 30 42 38 49 42 55

Here’s the data.

Can you see how the scatter diagram helps you visualize patterns in the data?

Can you see how this might help us to define the connection between open air

concert attendance and predicted number of hours sunshine for the day?

Hours sunshine goes

on the x-axis,

attendance on the

y-axis.

sunshine (hours)

attendance (100’s)

These are all the data points.

Sunshine is on the x-axis.

Attendance is

on the y-axis.

Concert Attendance and Sunshine

0 1 2 3 4 5 6 7 8

610 Chapter 15

sunshine (hours)

attendance (100’s)

Concert Attendance and Sunshine

0 1 2 3 4 5 6 7 8

We know we haven’t shown you how to analyze bivariate data yet,

but see how far you get in analyzing the scatter diagram for the

concert organizers.

What sort of patterns do you see in the chart? How can you relate

this to the underlying data? What do you expect open air concert

attendance to be like if it’s sunny? What about if it’s overcast?

sharpen your pencil