The Manga Guide to Statistics

Foowing

along so

far?

That's

true.

I hope you now understand

why "the cramer's coeicient

for the population is 'almost'

zero" is not the only nu

hypothesis, and there exist

some nu hypotheses that

sm diicult to prove, such

as "the cramer's coeicient

for the population is zero."

Also, Did You Notice That

Nu Hypotheses Are Positive

and use Phrases Like "...is..."

Or "...are The Same"?

alternative hypotheses are

negative, with phrases like

"...is not..." Or "...are not

the same."

This is how you

should think about

this maer.

Then set a hypothesis

that is oosite to the

nu hypothesis as the

alternative hypothesis.

4. P-value And Procedure for

hypothesis Tests

Yes I do. So

many of them

are very

extreme.

That's

convincing.

Use a hypothesis

that sms diicult

to prove and is

positive for the

nu hypothesis.

174 Chapter 7

4. P-value And Procedure for

hypothesis Tests

When making a

conclusion in a

hypothesis test...

There are two

ways to make a

judgment.

You told me about the

first one, but not the

second.

WHAT'S the

P-VALUE?

Though there

are some

dierences

depending

on which

hypothesis

test you are

doing,

in tests of

independence,

the p-value...

...is a probability that

gives you a value of

χ

0

2

the

same as or greater

than what has bn

calculated in the case in

question, when the nu

hypothesis is true.

In the previous

example...

it is the

probability shown

here.

(1) Whether the test

statistic is in the critical

region

(2) Whether the P-value

is smaer than the

significance level

Let’s Explore the Hypothesis Tests 175

You mean the part

that’s shaded?

Until personal

computers became

popular, it was

awfuy diicult to

calculate the p-value.

For that reason,

they used to

use (1)...

Oh,

reay...

...For most of

the conclusions

in hypothesis

tests.

What about

now?

That was around the

begiing of the 1990s.

Stop! Your fake

girl voice is

disturbing!

The p-value can

be calculated

easily by using

Microsoft

excel and other

software. So (2)

is starting to be

used more often.

That's

satisf ying.

The proce for (2)

is dierent from

the proce I just

explained for (1),

so...

"The puet statistician

is going to do the

analysis once again!"

T h h!

176 Chapter 7

Step 6p

Determine whether or not the P-value corresponding to the test statistic obtained in step 5 is smaller

than the significance level.

Let’s Explore the Hypothesis Tests 177

Teacher

As mentioned before, you can calculate the P-value using Excel

(though this depends on what type of hypothesis test you are

doing). See page 208 for details.

Analyst

The significance level is 0.05. Since Pearson’s chi-square test

statistic (χ

0

2

, which is the test statistic in this case) is 8.0091, the

P-value is 0.0182.

0.0182 < 0.05

Thus the P-value is smaller.

Step 7p

If the P-value is smaller than the significance level in step 6p, you reject the null hypothesis. If not, you

fail to reject the null hypothesis.

178 Chapter 7

Teacher

Even if the P-value was smaller than the significance level, you

cannot really conclude that the alternative hypothesis is “absolutely”

correct in a hypothesis test. The only conclusion you can make is:

“I would like to say that the alternative hypothesis is ‘absolutely’

correct . . . but there is a (

a× 100)% possibility that the null hypoth-

esis is correct.”

Analyst

The P-value was smaller than the significance level. Therefore,

you conclude in favor of the alternative hypothesis, “the Cramer’s

coefficient for the population is greater than 0.”