Griffiths D. Head First Statistics

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using hypothesis tests

Step 2: Choose your test statistic

Now that you’ve determined exactly what it is you’re going to test, you

need some means of testing it. You can do this with a test statistic.

The test statistic is the statistic that you use to test your hypothesis. It’s

the statistic that’s most relevant to the test.

What’s the test statistic for SnoreCull?

In our hypothesis test, we want to test whether SnoreCull cures 90% of

people or more. To test this, we can look at the probability distribution

according to the drug company, and see whether the number of

successes in the sample is significant.

If we use X to represent the number of people cured in the sample, this

means that we can use X as our test statistic. There are 15 people in the

sample, and the probability of success according to the drug company

is 0.9. As X follows a binomial distribution, this means that the test

statistic is actually:

X ~ B(15, 0.9)

I’m confused. Why are

we saying the probability

of success is 0.9? Surely

we don’t know that yet.

We choose the test statistic according to H

, the null

hypothesis.

We need to test whether there is sufficient evidence against the null

hypothesis, and we do this by first assuming that H

is true. We then look

for evidence that contradicts H

. For the SnoreCull hypothesis test, we

assume that the probability of success is 0.9 unless there is strong evidence

against this being true.

To do this, we look at how likely it is for us to get the results we did,

assuming the probability of success is 0.9. In other words, we take the

results of the sample and examine the probability of getting that result.

We do this by finding a critical region.

This is the test statistic

for our hypothesis test.

Decide on the

hypothesis you’re going

to test

Choose your test

statistic

Determine the critical

region for your decision

Find the p-value of the

test statistic

See whether the

sample result is within

the critical region

Make your decision

You

are

Here

We came up with this

test statistic back

on page 524.

532 Chapter 13

If 90% of people in the sample had

been cured, we could easily have assumed

that the drug company’s claims are

correct.

The fewer people who are cured,

the more likely it is that the

drug company claims are wrong.

The critical region of a hypothesis test is the set of values that present the

most extreme evidence against the null hypothesis.

Let’s see how this works by taking another look at the doctor’s sample. If 90%

or more people had been cured, this would have been in line with the claims

made by the drug company. As the number of people cured decreases, the

more unlikely it becomes that the claims of the drug company are true.

Here’s the probability distribution:

Step 3: Determine the critical region

At what point can we reject the drug company claims?

The fewer people there are in the sample who are successfully cured by

SnoreCull, the stronger the evidence there is against the claims of the drug

company. The question is, at what point does the evidence become so strong

that we confidently reject the null hypothesis? At what point can we reject the

claim that SnoreCull cures 90% of snorers?

What we need is some way of indicating at what point we can reasonably

reject the null hypothesis, and we can do this by specifying a critical region.

If the number of snorers cured falls within the critical region, then we’ll say

there is sufficient evidence to reject the null hypothesis. If the number of

snorers cured falls outside the critical region, then we’ll accept that there isn’t

sufficient evidence to reject the null hypothesis, and we’ll accept the claims

of the drug company. We’ll call the cut off point for the critical region c, the

critical value.

So how do we choose the critical region?

11 12 13 1410

If the number of people cured by SnoreCull falls in

this area, then we don’t have sufficient evidence

to reject H

. We’re only checking whether there’s

evidence that p < 0.9.

If the number of people cured by SnoreCull falls

in this critical region, then we can safely reject

the claims of the drug company, H

. Our alternate

hypothesis is that p < 0.9, so we look to this region

for evidence.

Cutoff point,

or critical

value

Decide on the

hypothesis you’re going

to test

Choose your test

statistic

Determine the critical

region for your decision

Find the p-value of the

test statistic

See whether the

sample result is within

the critical region

Make your decision

You

are

Here

finding critical regions

you are here 4 533

using hypothesis tests

Before we can find the critical region of the hypothesis test, we first

need to decide on the significance level. The significance level of

a test is a measure of how unlikely you want the results of the sample

to be before you reject the null hypothesis H

. Just like the confidence

level for a confidence interval, the significance level is given as a

percentage.

As an example, suppose we want to test the claims of the drug

company at a 5% level of significance. This means that we choose

the critical region so that the probability of fewer than c snorers

being cured is less than 0.05. It’s the lowest 5% of the probability

distribution.

5% 95%

Critical region

If the number of snorers cured by

SnoreCull falls in the critical region, then

we’ll reject the null hypothesis.

The significance level is normally represented by the Greek letter α. The

lower α is, the more unlikely the results in your sample need to be before

we reject H

So what significance level should we use?

Let’s use a significance level of 5% in our hypothesis test. This

means that if the number of snorers cured in the sample

is in the lowest 5% of the probability distribution, then we

will reject the claims of the drug company. If the number of

snorers cured lies in the top 95% of the probability distribution,

then we’ll decide there isn’t enough evidence to reject the null

hypothesis, and accept the claims of the drug company.

If we use X to represent the number of snorers cured, then we

define the critical region as being values such that

P(X < c) < α

where

α = 5%

Significance level

The significance level is

represented by α. It’s a

way of saying how unlikely

you want your results to be

before you’ll reject H

Vital StatisticsVital Statistics

To find the critical region, first decide on the significance level

If H

is true, we are 95% certain that the number

of snorers cured will fall within this region

534 Chapter 13

Critical Regions Up Close

When you’re constructing a critical region for your test, another thing you need to

be aware of is whether you’re conducting a one-tailed or two-tailed test. Let’s

look at the difference between the two, and what impact this has on the critical

region?

One-tailed tests

A one-tailed test is where the critical region falls at one end

of the possible set of values in your test. You choose the level

of the test—represented by α—and then make sure that the

critical region reflects this as a corresponding probability.

The tail can be at either end of the set of possible values, and

the end you use depends on your alternate hypothesis H

If your alternate hypothesis includes a < sign, then use the

lower tail, where the critical region is at the lower end of the

data.

If your alternate hypothesis includes a > sign, then use the

upper tail, where the critical region is at the upper end of

the data.

We’re using a one-tailed test for the SnoreCull hypothesis

test with the critical region in the lower tail, as our alternate

hypothesis is that p < 0.9.

100% - α

A one-tailed test

at the α level

Here we’re using the lower tail.

100% - α

The critical region this

time is in the upper tail,

still at the α level.

Two-tailed tests

A two-tailed test is where the critical region is split over both

ends of the set of values. You choose the level of the test α,

and then make sure that the overall critical region reflects this

as a corresponding probability by splitting it into two. Both

ends contain α/2, so that the total is α.

You can tell if you need to use a two-tailed test by looking

at the alternate hypothesis H

. If H

contains a ≠ sign, then

you need to use a two-tailed test as you are looking for some

change in the parameter, rather than an increase or decrease.

We would have used a two-tailed test for our SnoreCull if our

alternate hypothesis had been p ≠ 0.9. We would have had to

check whether significantly more or significantly fewer than

90% of patients had been cured

100% - α

α/2

This is a two-tailed test, where

the critical region is split over

the two tails.

critical regions in depth

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using hypothesis tests

Step 4: Find the p-value

Now that we’ve looked at critical regions, we can move on to step 4, finding

the p-value.

A p-value is the probability of getting a value up to and including the one

in your sample in the direction of your critical region. It’s a way of taking

your sample and working out whether the result falls within the critical

region for your hypothesis test. In other words, we use the p-value to say

whether or not we can reject the null hypothesis.

How do we find the p-value?

How we find the p-value depends on our critical region and our test statistic.

For the SnoreCull test, 11 people were cured, and our critical region is

the lower tail of the distribution. This means that our p-value is P(X 11),

where X is the distribution for the number of people cured in the sample.

As the significance level of our test is 5%, this means that if P(X 11) is

less than 0.05, then the value 11 falls within the critical region, and we can

reject the null hypothesis.

0.05 0.95

If P(X ≤ 11) is less than 0.05, then that means that 11 is

inside the critical region, and we can reject H

We know from step 2 that X ~ B(15, 0.9). What’s P(X ≤ 11)?

Decide on the

hypothesis you’re going

to test

Choose your test

statistic

Determine the critical

region for your decision

Find the p-value of the

test statistic

See whether the

sample result is within

the critical region

Make your decision

You

are

Here

536 Chapter 13

We know from step 2 that X ~ B(15, 0.9). What’s P(X ≤ 11)? Is 11

inside or outside the critical region?

P(X ≤ 11) = 1 - P(X ≥ 12)

= 1 - (

x0.1

x0.9

x0.1

x0.9

x0.1x0.9

+ 0.9

)

= 1 - (0.1285 + 0.2669 + 0.3432 + 0.2059)

= 1 - 0.9445

= 0.0555

We’ve found the p-value

To find the p-value of our hypothesis test, we had to find P(X 11). This

means that the p-value is 0.0555.

Do I always calculate

p-values in the same way?

What if my critical region

had been the upper tail?

A p-value is the probability of getting the results in the

sample, or something more extreme, in the direction of

the critical region.

In our hypothesis test for SnoreCull, the critical region is the lower tail of

the probability distribution. In order to see whether 11 people being cured

of snoring is in the critical region, we calculated P(X 11), as this is the

probability of getting a result at least as extreme as the results of our sample

in the direction of the lower tail.

0.05 0.95

We want to find whether 11 people being cured is in the

critical region here, so we use P(X ≤ 11) to evaluate this.

Had our critical region been the upper tail of the probability distribution

instead, we would have needed to find P(X 11). We would have counted

more extreme results as being greater than 11, as these would have been

closer to the critical region.

sharpen solution

= 1, and so does 0.1

so we’re just left with 0.9

you are here 4 537

using hypothesis tests

nore

ull

nore

ull

48 T

ableTS

Now that we’ve found the p-value, we can use it to see whether the result

from our sample falls within the critical region. If it does, then we’ll have

sufficient evidence to reject the claims of the drug company.

Our critical region is the lower tail of the probability distribution, and

we’re using a significance level of 5%. This means that we can reject the

null hypothesis if our p-value is less that 0.05. As our p-value is 0.0555, this

means that the number of people cured by SnoreCull in the sample doesn’t

fall within the critical region.

I win!

5% 95%

This is the

critical region.

The p-value is 0.056, so it’s

just outside the critical region.

Step 6: Make your decision

We’ve now reached the final step of the hypothesis test. We can decide

whether to accept the null hypothesis, or reject it in favor of the alternative.

The p-value of the hypothesis test falls just outside the critical region of

the test. This means that there isn’t sufficient evidence to reject the null

hypothesis. In other words:

We accept the claims of the drug company

Step 5: Is the sample result in the critical region?

Decide on the

hypothesis you’re going

to test

Choose your test

statistic

Determine the critical

region for your decision

Find the p-value of the

test statistic

See whether the

sample result is within

the critical region

Make your decision

You

are

Here

Decide on the

hypothesis you’re going

to test

Choose your test

statistic

Determine the critical

region for your decision

Find the p-value of the

test statistic

See whether the

sample result is within

the critical region

Make your decision

You

are

Here

538 Chapter 13

But those results

aren’t what the doctor

wants. Can’t we test at a

different level?

Once you’ve fixed the significance level of the test,

you can’t change it.

The test needs to be completely impartial. This means that you decide

what level you need the test to be at, based on what level of evidence

you require, before you look at what evidence you actually have.

If you were to look at the amount of evidence you have before deciding

on the level of the test, this could influence any decisions you made.

You might be tempted to decide on a specific level of test just to get the

result you want. This would make the outcome of the test biased, and

you might make the wrong decision.

So what did we just do?

Let’s summarize what we just did.

First of all, we took the claims of the drug company, which the doctor had

misgivings about. We used these claims as the basis of a hypothesis test. We

formed a null hypothesis that the probability of curing a patient is 0.9, and

then we applied this to the number of people in the doctors sample.

We then decided to conduct a test at the 5% level, using the success rate in

the doctor’s sample. We looked at the probability of 11 people or fewer being

cured, and checked to see whether the probability of this was less than 5%,

or 0.05. In other words, we looked at the probability of getting a result this

extreme, or even more so.

Finally, we found that at the 5% level, there wasn’t strong enough evidence to

reject the claims of the drug company.

hypothesis testing in review

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using hypothesis tests

What significance level should I normally test at?

A: It all depends how strong you want the evidence to be before

you reject the null hypothesis. The stronger you want the evidence to

be, the lower your significance level needs to be.

The most common significance level is 5%, although you sometimes

see tests at the 1% level. Testing at the 1% means that you require

stronger evidence than if you test at the 5% level.

Does the significance level have anything in common with

the level of confidence for confidence intervals?

A: Yes, they have0 a lot in common. When you construct a

confidence interval for a population parameter, you want to have

a certain degree of confidence that the population parameter lies

between two limits. As an example, if you have a 95% level of

confidence, this means that the probability that the population

parameter lies between the two limits is 0.95.

The level of significance reflects the probability that values will lie

outside a certain limit. As an example, a significance level of 5%

means that your critical region must have a probability of 0.05.

I still have doubts.

I wonder what would

happen if I took a

larger sample...

In a hypothesis test, you take a claim and test it

against statistical evidence.

The claim that you’re testing is called the null

hypothesis test. It’s represented as H

, and it’s the

claim that’s accepted unless there’s strong statistical

evidence against it.

The alternate hypothesis is the claim we’ll accept

if there’s strong enough evidence against H

. It’s

represented by H

The test statistic is the statistic you use to test your

hypothesis. It’s the statistic that’s most relevant to

the test. You choose the test statistic by assuming

that H

is true.

The significance level is represented by α. It’s a way

of saying how unlikely you want your results to be

before you’ll reject H



The critical region is the set of values that presents

the most extreme evidence against the null

hypothesis test. You choose your critical region by

considering the significance level and how many tails

you need to use.

A one-tailed test is when your critical region lies

in either the upper or the lower tail of the data. A

two-tailed test is when it’s split over both ends.

You choose your tail by looking at your alternate

hypothesis.

A p-value is the probability of getting the result

of your sample, or a result more extreme in the

direction of your critical region.

If the p-value lies in the critical region, you have

sufficient reason to reject your null hypothesis. If

your p-value lies outside your critical region, you

have insufficient evidence.



540 Chapter 13

What if the sample size is larger?

So far the doctor has conducted her trial using a sample of just 15 people,

and on the basis of this, there was insufficient evidence to reject the

claims of the drug company.

It’s possible that the size of the sample wasn’t large enough to get an

accurate result. The doctor might get more reliable results by using a

larger sample.

Here are the results from the doctor’s new trial:

What’s the null hypothesis of this new problem?

What’s the alternate hypothesis?

We want to determine whether the new data will

make a difference in the outcome of the test.

Let’s run through another hypothesis test, this time with the larger

sample.

Cured? Yes No

Frequency

80 20

I want to conduct a new

hypothesis test using

these new results.

a new sample for hypothesis testing