Griffiths D. Head First Statistics

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using hypothesis tests

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SnoreCull failed the test

This time when we performed a hypothesis test on SnoreCull,

there was sufficient evidence to reject the null hypothesis.

In other words, we can reject the claims made by the drug

company.

Busted...

Hypothesis tests require evidence.

With a hypothesis test, you accept a claim and then put it on trial. You

only reject it if there’s enough evidence against it. This means that the

tests are impartial, as you only make a decision based on whether or

not there’s sufficient evidence.

If we had just accepted the doctor’s opinion in the first place, we

wouldn’t have properly considered the evidence. We would have

made a decision without considering whether the results could have

been explained away by mere coincidence. As it is, we have enough

evidence to show that the results of the sample are extreme enough

to justify rejecting the null hypothesis. The results are statistically

significant, as they’re unlikely to have happened by chance.

So does this guarantee that the claims of the drug company

are wrong?

Shouldn’t we have

just accepted the doctor’s

opinion in the first place?

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Of course it is. We’ve

done a hypothesis test, and

we’ve used it to prove that

the drug company is lying.

Mistakes can happen

So far we’ve looked at how we can use the results of a sample as evidence in

a hypothesis test. If the evidence is sufficiently strong, then we can use it to

justify rejecting the null hypothesis.

We’ve found that there is strong evidence that the claims of the drug

company are wrong, but is this guaranteed?

Even though the evidence is strong, we can’t absolutely

guarantee that the drug company claims are wrong.

Even though it’s unlikely, we could still have made the wrong decision. We can

examine evidence with a hypothesis, and we can specify how certain we want

to be before rejecting the null hypothesis, but it doesn’t prove with absolute

certainty that our decision is right.

The question is, how do we know?

Conducting a hypothesis test is a bit like putting a prisoner on trial in front

of a jury. The jury assumes that the prisoner is innocent unless there is strong

evidence against him, but even considering the evidence, it’s still possible for

the jury to make wrong decisions. Have a go at the exercise on the next page,

and you’ll see how.

How can we make the wrong decision if we’re conducting

a hypothesis test? Don’t we do a hypothesis test to make sure

we don’t?

A: When you conduct a hypothesis test, you can only make a

decision based on the evidence that you have. Your evidence is

based on sample data, so if the sample is biased, you may make the

wrong decision based on biased data.

I’ve heard of something called significance tests. What are

they?

A: Some people call hypothesis tests significance tests. This is

because you test at a certain level of significance.

our hypothesis might still be wrong

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using hypothesis tests

A prisoner is on trial for a crime, and you’re on the jury. The jury’s

task is to assume the prisoner is innocent, but if there’s enough

evidence against him, they need to convict him.

1. In the trial, what’s the null hypothesis?

2. What’s the alternate hypothesis?

3. In what ways can the jury make a verdict that’s correct?

4. In what ways can the jury make a verdict that’s incorrect?

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A prisoner is on trial for a crime, and you’re on the jury. The jury’s

task is to assume the prisoner is innocent, but if there’s enough

evidence against him, they need to convict him.

1. In the trial, what’s the null hypothesis?

2. What’s the alternate hypothesis?

3. In what ways can the jury make a verdict that’s correct?

4. In what ways can the jury make a verdict that’s incorrect?

The null hypothesis is that the prisoner is innocent, as that is what we have to

assume until there’s proof otherwise.

The alternate hypothesis is that the prisoner is guilty. In other words, if there’s sufficient proof

that the prisoner is not innocent, then we’ll accept that he’s guilty and convict him.

We can make a correct verdict if:

The pr

isoner is innocent, and we find him innocent.

The prisoner is guilty, and we find him guilty.

We can make an incorrect verdict if

The pr

isoner is innocent, and we find him guilty.

The prisoner is guilty, and we find him innocent.

sharpen solution

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using hypothesis tests

The errors we can make when conducting a hypothesis

test are the same sort of errors we could make when

putting a prisoner on trial.

Hypothesis tests are basically tests where you take a claim and put it on trial

by assessing the evidence against it. If there’s sufficient evidence against it,

you reject it, but if there’s insufficient evidence against it, you accept it.

You may correctly accept or reject the null hypothesis, but even considering

the evidence, it’s also possible to make an error. You may reject a valid null

hypothesis, or you might accept it when it’s actually false.

Statisticians have special names for these types of errors. A Type I error is

when you wrongly reject a true null hypothesis, and a Type II error is when

you wrongly accept a false null hypothesis.

The power of a hypothesis test is the probability that that you will correctly

reject a false null hypothesis.

So what does

putting prisoners on

trial have to do with

hypothesis testing?

Accept H

Reject H

True

Type I error

False

Type II error

Decision from hypothesis test

Actual

situation

This gives you the

power of your test.

These are both

types of errors

How do you think we can find the probability of making a Type I error? How do

you think we can find the probability of making a Type II error?

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Let’s start with Type I errors

A Type I error is what you get when you reject the null hypothesis when

the null hypothesis is actually correct. It’s like putting a prisoner on trial

and finding him guilty when he’s actually innocent.

But I’m

innocent

A Type I error is when you reject

when actually it’s correct.

So what’s the probability of

getting a Type I error?

If you get a Type I error, then this means that

the null hypothesis must have been rejected.

In order for the null hypothesis to have been

rejected, the results of your sample must be in

the critical region.

The probability of getting a Type I error is the probability of your

results being in the critical region. As the critical region is defined by the

significance level of the test, this means that if the significance level of

your test is α, the probability of getting a Type I error must be also be α.

In other words,

If you get a Type I error,

your test statistic must be

here in the critical region.

P(Type I error) = α

where α is the significance level of the test.

type i and type ii errors

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What about Type II errors?

A Type II error is what you get when you accept the null hypothesis,

and the null hypothesis is actually wrong. It’s like putting a prisoner on

trial and finding him innocent when he’s actually guilty.

Got away

with it

A Type II error is when you accept

when actually it’s wrong.

The probability of getting a Type II error is normally represented by

the Greek letter β.

P(Type II error) = β

So how do we find β?

Finding the probability of a Type II error is more difficult than finding

the probability of getting a Type I error. Here are the steps that are

involved, and we’ll show you how to go through them on the next page.

Check that you have a specific value for H

Without this, you can’t calculate the probability of getting a Type II error.

Find the range of values outside the critical region of your test.

If your test statistic has been standardized, the range of values must be de-standardized.

Find the probability of getting this range of values, assuming H

is true.

In other words, we find the probability of getting the range of values outside the critical

region, but this time, using the test statistic described by H

rather than H

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Finding errors for SnoreCull

Let’s see if we can find the probability of getting Type I and Type II errors for

the SnoreCull hypothesis test. As a reminder, our standardized test statistic is

Z = X - 90

where X is the number of people cured in the sample. The significance level of

the test is 5%.

Let’s start with the Type I error

A Type I error is what you get when you reject the null hypothesis when

actually it’s true. The probability of getting this sort of error is the same as the

significance level of the test, so this means that

P(Type I error) = 0.05

So what about the Type II error?

A Type II error is what you get when you accept the null hypothesis when the

alternate hypothesis is true. We can only calculate this if H

specifies a single

specific value, so let’s use an alternate hypothesis of p = 0.8, as this is the

proportion of successes in the doctor’s sample. This means that our hypotheses

become

: p = 0.9

: p = 0.8

The reason why H

must specify an exact value for p is so that we can calculate

probabilities using it. If we used an alternate hypothesis of p < 0.9, we wouldn’t

be able to use it to calculate the probability of getting a Type II error.

If you need to calculate the

probability of getting a Type II error

in an exam, you’ll be given H

This means that you won’t have to decide on the

alternate hypothesis yourself. If you need to calculate

this sort of error, it will be given to you.

This time we’ll use H

: p = 0.8 instead of H

: p < 0.8. We can only

calculate the probability of getting a Type II error if we have a

single specific value for the alternate hypothesis.

This gives you the probability of rejecting

the null hypothesis that 90% of people are

cured when it’s true.

calculating type i and type ii errors

To look up probabilities using the

alternate hypothesis probability

distribution, we need an exact

value for p.

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These values are inside

the critical region.

Z ~ N(0, 1)

We need to find the range of values

Now that the alternate hypothesis H

gives a specific value for p, we can move

on to the next step. We need to find the values of X that lie outside the critical

region of the hypothesis test.

We saw back on page 548 that the critical region for the test is given by Z

< -1.64—in other words, P(Z < -1.64) = 0.05. This means that values that fall

outside the critical region are given by Z ≥ -1.64.

Values here are outside

the critical region.

If we de-standardize this, we get

X - 90 ≥ -1.64

X - 90 ≥ -1.64 x 3

X ≥ -4.92 + 90

X ≥ 85.08

In other words, we would have accepted the null hypothesis if 85.08 people

or more had been cured by SnoreCull.

The final thing we need to do is work out P(X ≥ 85.08), assuming that H

is true. That way, we’ll be able to work out the probability of accepting the

null hypothesis when actually H

is true instead. As we’re using the normal

distribution to approximate X, we need to use a probability distribution

X ~ N(np, npq), where n = 100 and p = 0.8. This gives us

X ~ N(80, 16)

This means that if we can calculate P(X ≥ 85.08) where X ~ N(80, 16), we’ll

have found the probability of getting a Type II error.

We calculate this in the same way we calculate other normal distribution

probabilities, by finding the standard score and then looking up the value in

standard normal probability tables.

-1.64

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This is the usual way of calculating the

standard score; just subtract the expectation,

and divide by the standard deviation.

P(Type II error) = 0.102

Why is it so much harder to find P(Type II error) than

P(Type I error)?

A: It’s because of the way they’re defined. A Type I error is what

you get when you wrongly reject the null hypothesis. The probability

of getting this sort of error is the same as α, the significance level of

the test.

A Type II error is the error you get when you accept the null

hypothesis when actually the alternate hypothesis is true. To find the

probability of getting this sort of error, you need to start by finding

the range of values in your sample that would mean you accept the

null hypothesis. Once you’ve found these values, you then have to

calculate the probability of getting them assuming that H

is true.

Do I need to use the normal distribution every time I want

to find the probability of getting a Type II error?

A: The probability distribution you use all depends on your test

statistic. In this case, our test statistic followed a normal distribution,

so that’s the distribution we used to find P(Type II error). If our test

statistic had followed, say, a Poisson distribution, we would have

used a Poisson distribution instead.

This gives you the probability of accepting

the null hypothesis that 90% of people are

cured when actually 80% of people are.