Flashcards in L7 - Test of Hypothesis Deck (16)

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1

## What is the logic behind the test of hypothesis?

###
- we do not know what µ is (all we have is X(bar) )

- how likely is it that µ is equal to a given number?

- Go back to the example of the mobile phone company; suppose that the

manager claims that their average monthly bill, µ, is £31;

- Your sample gives X(bar) = 32: does it mean that the manager is lying?

Claim: µ= 31 --- Sample evidence: X(bar) = 32

-Is the claim credible? Is it supported by sample evidence?

- this is a hypothesis

- We ask: is it likely for the average bill to be 31 given that the sample

average is 32? Or viceversa, how likely would I be to get a sample value

of 32 if the true mean were 31?

- This is the idea behind hypothesis testing: you have a value in mind (the

claim, or hypothesis: 31) and you want to test it against sample evidence

(32) and form an opinion on whether it is credible or not.

- More extreme example: if the claim was µ = 8 and yet X(bar)= 32 you

would probably suspect the claim (the hypothesis) not to be true;

Why?

- Because the sample mean is more likely than not to be close to the true

mean (unbiased estimator), and 32 seems very far from 8 (unless the

variance is really huge). You’d be thinking how likely would it be to get a

sample average of 32 if the true mean were 8?

2

## Where do you start with the hypothesis testing procedure?

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- Testing a hypothesis about a parameter, for instance the mean µ of a variable, means that we have a value in mind and we want to check whether/how much the sample evidence supports it.

- The value we test (µ{0}) is called the null hypothesis, labelled H{0}

H{0}: µ = µ{0}

eg: H{0}: µ = 8

- Together with the null we have to specify an alternative hypothesis

H{1}; this is a non-specific statement that holds true if our null is rejected

by the data.

- There are three kinds of alternative hypotheses:

3

## What are the 3 kinds of alternative hypotheses?

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- composite (or open) alternative

- One sided alternative - right tail;

- One sided alternative - left tail

- The choice depends on our assumptions and previous knowledge about the

problem, and should be made before looking at the sample data.

4

## What is a Composite (or open) alternative?

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- Starting with H{0}: µ = µ{0}

- then H{1}: µ ≠ µ{0}

- therefore µ could be > µ{0} or < µ{0}

5

## What is a One sided alternative - right tail?

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- Starting with H{0}: µ = µ{0}

- then H{1}: µ > µ{0}

- therefore if not µ{0} then µ must be > µ{0}

6

## What is a One sided alternative - left tail?

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- Starting with H{0}: µ = µ{0}

- then H{1}: µ < µ{0}

- therefore if not µ{0} then µ must be < µ{0}

7

## What is the Hypothesis testing procedure for a composite (or open) alternative?

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- Suppose we are testing:

H{0}:µ = µ{0} vs H{1}: µ ≠ µ{0}

- We now choose a significance level α (the area in the tails) for our test,

say of 5%.

- This means that we can construct a 95% confidence interval around the

hypothesised mean of µ{0} so that if our hypothesis is true then this is also

true

- P{ -1.96 < [X(bar)-µ{0}/ sqrt(σ^2/n)] < +1.96] = 0.95

- If µ = µ{0} then 95% of the values of must lie within this range

- We now look at where our sample mean falls:

- If it falls in the 95% area (acceptance region) we conclude that there is

enough evidence to support the null hypothesis and we do not reject

the null hypothesis.

- If it falls in one of the tails (the rejection region) we reject the null hypothesis: with a mean µ{0} it would be very unlikely (less than 5%) to get a sample average equal to X(bar) , i.e. our sample evidence does not support the null hypothesis.

- Alternatively we can use the C.I. to check where the null falls: centre the distribution around , i.e. around the sample evidence, and look how far from it is the null hypothesis.

- The inference will be exactly the same

8

## What is the fastest way to carry out the hypothesis test for a composite (or open) alternative?

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- A simpler, faster way to carry out this test is to compute formula (f1) directly rather than building a confidence interval. Recall:

- P{ -1.96 < [X(bar)-µ{0}/ sqrt(σ^2/n)] < +1.96] = 0.95

- We can compute directly the ratio X(bar) -µ{0}/ sqrt(σ^2/n)] and check if the resulting value fall withing +/- 1.96

9

## What are the three different ways of testing a hypothesis?

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- centering around µ

- centering around X(bar)

- testing whether its in between the values of Z on the confidence interval

10

## What is the significance level?

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- The level of significance = α = area in the tails = the probability of an

error-type I.

- An error-type I is the error of rejecting a true null hypothesis.

- We reject a hypothesis because we consider it unlikely; unlikely is not

the same as completely impossible. All we are saying is that if the null

were true then our sample mean would have a probability of occurring

of α or less.

- The smaller are the tails thus the lower the chance of rejecting the null

and therefore of making a type I error

11

## It is a good idea to keep the tails as small as possible?

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-No, because in this way you increase the probability of an error type

II: the probability of not rejecting a false null hypothesis:

- i.e., the true mean is not 0

, however the test statistic falls in the

acceptance region and we wrongly conclude in favour of the null

hypothesis. This error is larger the larger is our acceptance area.

- The probability of a type II error is denoted β.

- We define the robustness of a test as 1- β : a test is robust the smaller

is the probability of a type II error i.e. the less likely you are to accept a

false null.

12

## What does a large significance level show?

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- Large significance level (large tails) => larger probability of error-type I

(reject a true hypothesis)

13

## What does a small significance level show?

### Small tails, i.e. larger acceptance region => larger probability of errortype II (accept a wrong hypothesis)

14

## How do you test against a one-sided alternative?

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- Very simply we just use only one tail for the whole significance level (for

instance the whole of the 5%), then proceed as before:

- if our test value falls in the tail we reject, if not we do not reject.

- The tail we use depends on the alternative hypothesis: given H{0}: μ = μ{0}

- H{1}: μ > μ{0} we use the right tail

- we reject if X(bar)-µ{0}/ sqrt(σ^2/n)] > z and do not reject if it is smaller

- H{1}: μ < μ{0 we use the left tail

- we reject if X(bar)-µ{0}/ sqrt(σ^2/n)] < z and do not reject if it is larger

15

## What does a one-sided alternative test mean?

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- Ex right tailed test, H{1}: µ > µ{0}

- Why? Because by saying that if not µ{0} the true mean µ must be larger

than it we are saying that the true distribution lies to the right of

distribution under the null. Vice versa for the left tail test.

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