Sections B,C of ECO252 Outline

252onesl 9/20/07 (Open this document in 'Outline' view!)

B. HYPOTHESIS TESTS FOR ONE SAMPLE

1. The Meaning of Hypothesis Testing

A hypothesis is a statement about the characteristics of a population.

To be of any use to us it must be quantifiable and testable.

The hypothesis to be tested is usually called the Null Hypothesis .

A rival hypothesis is called the Alternative or Research Hypothesis

or . It is usually true that the null hypothesis will be a hypothesis

of "no difference" and the alternative hypothesis covers all other

possibilities.

To start out ask "What do I want to know about a population or

populations?" Can I state this in terms of population parameters?

Can I state this in terms of a testable hypothesis (null hypothesis)?

Does my null hypothesis say that these parameters or differences

between these parameters are insignificant (i.e. not distinct from

zero)?

Next ask "What am I assuming about the population or populations?"

Are the parameters that I am testing appropriate to the type of

population that I am assuming? What can I use to test my

hypothesis? Can I find a sample statistic or statistics to do the job ?

How many samples do I need? Can I calculate the sample statistic

or statistics.? What distribution does the test statistic have? Is this

in accord with the null hypothesis? What errors am I likely to make?

Usually there are three approaches to hypothesis testing

involving a statement about a parameter of a population:

(i) The Test Ratio Method, in which a ratio involving an

estimate of a parameter is tested against a well known

distribution like t;

(ii) The Critical Value Method, in which values of

estimates of a parameter are found which could lead to

rejection of ;

and

(iii) The Confidence Interval Method, in which a

confidence interval is constructed for the parameter and

compared to the value in the null hypothesis.

If I use a test ratio, what is the probability of getting values as

extreme or more extreme than I actually got? (This probability is

the p-value: the lower it is the less likely it is that the null

hypothesis is true. If the p-value falls below the significance level,

I can say that I reject the null hypothesis.)

If I use a test statistic and my significance level is 5% or 1%,

did the value fall among the most likely 95% or 99% of values? Or

was it a very unlikely value?

If I use a confidence interval, did the parameter value in my

null hypothesis fall in the confidence interval?

Remember:

a. A null hypothesis is usually a statement

about a parameter of a population. It is

never a statement about a sample statistic.

A sample statistic is used to test the

hypothesis.

b. A null hypothesis usually contains an

equality, an alternate hypothesis does not

contain an equality.

c. A null hypothesis often says that a

parameter or a difference between parameters

is insignificant. If a result is significant we

reject the null hypothesis.

2. Steps for Testing a Hypothesis Applied to testing

for a Population Mean

a.Outline

i. State the problem as two hypotheses.

ii. Quantify the hypotheses.

iii. Identify the statistic, ratio or interval to be used

iv. Determine the sampling distribution of the statistic to

be used.

v. Select a level of significance.

vi. Find a value or values of the test ratio or statistic that

would lead to rejection of the null hypothesis.

vii. Compute a value of the statistic or ratio from a

random sample

viii. By comparing the results of (vii) with the values found

in (vi) reject or do not reject (‘accept’) the null hypothesis.

b. Application to a Population Mean

To test against . Assume that we

have computed from our sample, and that we do not know

i. Test Ratio:

ii. Critical Value:

iii. Confidence Interval:

Note: If , the population standard deviation, is

known, replace and with and .

c. One-sided Tests.

To test against or against

, if you use a critical value or a confidence

interval you must use a one-sided one. Replace with .

One-sided tests take more thinking than two-sided test, and the

most common error is in stating the null hypothesis. In a problem

statement, the question asked is often the alternative hypothesis,

not the null. Always ask yourself if the statement contains a strict

inequality. If it does it cannot be a null hypothesis.

Examples:

Question: Is the mean income less than 20000?

Question: Is the mean income at least 20000?

Question: Is the mean income more than 20000?

Question: Is the mean income at most 20000?

3. The Use of p-value Instead of Significance Levels.

A p-value is a measure of the credibility of the null hypothesis and is

defined as the probability that a test statistic or ratio as as

or than the observed statistic or ratio could occur,

assuming that the null hypothesis is true.

Note: If we have a p-value and want to do a conventional hypothesis

test, we can reject the null hypothesis if the p-value is below the

significance level. The p-value can thus be said to represent the

smallest level of significance at which the null hypothesis can be

rejected.

Other interpretations are:

a) (i) If we strongly doubt ,

(ii) If we somewhat doubt , and

(iii) If we cannot doubt ; or

b) (i) If results are very significant,

(ii) If results are significant,

(iii) If results are marginally significant

and

(iv) if results are not significant.

For an example using t see 252onesx0. For an example using z replace t with z in this paragraph and see 252doctor

4. Type One and Type Two Errors

a. Definitions

A Type one error is rejecting when is true.

A Type two error is not rejecting when is false.

/ / /
/

Do not reject

Not an error

Type II Error

Reject

Type I error

Not an error

b. Probabilities

/ / /
/ / /
/ / /

5. Hypotheses about a Proportion

a. Tests: (For an example see 252onesx1)

To test against

i. Test Ratio:

ii. Critical Value:

iii. Confidence Interval:

(b. Continuity Correction.

The continuity correction acts to expand the 'accept' interval

by in each direction. It should be used if .

i. Test Ratio:

This is the same as testing

against

ii. Critical Value:

iii. Confidence Interval: )

6. The Sign Test

a. The Sign Test for a Median.

To test against

In any distribution outside of the normal distribution, it is

usually easier to use the p-value approach. For example, let

us assume that we are testing the hypotheses

and , where  is, as before, the median. The

most important fact to know about testing for a median is that

numbers above and below a median are equally likely to

occur in a random sample.

A test of the median is a test of the proportion of points above

or below the alleged median.

So let us use as the proportion of the observations

in our population that are above 25. ( could just as easily be

the proportion below 25.) If this is true, and we are working

with a continuous distribution, our hypotheses become

and . Now let us assume that we

take a sample of and that we find that , the number

of points above 25, is 5. We expect that half of our points,

or 10, will be above 25, so 5 seems low. We thus use a

binomial table to find for

The table tells us that . Since this is a

two-sided test, we double this probability to .0414 and use

this as our p-value. If our confidence level is 95%, our

significance level must be 5%, and, since the p-value is below

5%, we reject the null hypothesis.

(But if we are to repeat tests, it may be wise to

define acceptance and rejection regions by defining two

critical values, and saying that if , we

will reject the null hypothesis. Again assume that the

significance level is .

We can use the p-value approach by saying that, if we

would reject the null hypothesis using the p-value approach

for some value of , that value is in our rejection region.

Starting from the bottom, try . From the table for

. Since this p-value

is below , we would reject if x were 0. We come to a

similar conclusion if takes values of 2, 3 or 4. If ,

we have already seen that , and that we

would still reject the null hypothesis. But if we try ,

we find which is above , so we

‘accept’ if x is 6 or larger.

But, since this is a two-sided test, it is also possible that is

too large. For example if x is 16,

= 1 - .9941.

Since this is below , we would reject the null hypothesis if

were 16 or larger. So try . . This is still too

low for acceptance, so try 14. . Since this is above , we would

not reject the null hypothesis if were 14. We can thus say

that we do not reject the null hypothesis if is between 6 and

14, or that our critical values for are 5 and 15. If we now

look back at the cumulative binomial table, we see that we

rejected the null hypothesis for probabilities below .025

and above .975 .) {bin}

Let's try a one-sided problem. Suppose that our null

hypothesis is that median income in a region is at least $20000

and that we take a sample with the results shown below.

Let .

Our hypotheses are . Let / Index / Income
be the proportion of numbers in the population below / 1 / 10132
20000. If the median is exactly 20000, will be exactly / 2 / 11252
0.5. But if the median is above 20000, will be below .5. / 3 / 13475
We can replace our original hypotheses with and / 4 / 14260
. . We see that , the quantity of numbers in the / 5 / 16871
sample below 20000, is 7. Our expected number of items / 6 / 19357
below 20000 is , so 7 is high and our p- / 7 / 19438
value will be / 8 / 23010
. Since the p-value is above the significance level, / 9 / 30278
we accept the null hypothesis. / 10 / 35932

(If we wish to set up accept and reject zones for this

one sided test, we need to try higher values of . A value of

is still too small; it gives a probability of .0547, which

is above , so try . According to the binomial table for

, =.0107,

which is below . So 9 is our critical value, and we will

reject the hypothesis if .)

To clarify the correspondence between hypotheses about a

median and hypotheses about a proportion, let us assume that

is the proportion of the data above 20000. If 2000 is the

median, then, by definition of the median, is one half. But

let us assume that the median is above 2000, say 2100. Then

one half of the data must be above 2100, so that more than one

half of the data must be above 2000, which means less than one

half of the data is below 2000. Since a hypothesis about a

median is a hypothesis about a proportion,

corresponds to . The table below shows these

correspondences depending on the definition of .

Hypotheses aboutHypotheses about a proportion

a median If is the If is the

proportion aboveproportion below

b. The Sign Test more Generally.

This technique can be used in other ways. For

instance let us say that we wish to check the effectiveness of a

product brochure. A sample of 17 clients is asked about their

impression of a product. Then they read the brochure and once

again are asked their impression. We write a (+) if their

impression has improved and a () if it is worse. A zero

indicates no change. Our results are as follows:

Client / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15 / 16 / 17
Sign / + / + / + / + / 0 /  / + / 0 /  / + / + / + /  / + / 0 / + / +

Since we are hoping for a positive effect, count the zeros as

minuses. We will use the brochure if we believe that the

majority of the population will respond favorably. Let p be the

proportion of plusses in the population. Our hypotheses are

. There are 11 plusses so that we must

find when and . A large binomial table

says that this value is .166, so that we must accept the null

hypothesis and not use the brochure.

In the absence of a binomial table we must use the

Normal approximation to the binomial distribution. If

is our observed proportion, we use . But for the

sign test, . So

(For relatively small values of , a continuity correction is

advisable, so try , where the + applies if

, and the  applies if . In the problem above ,

where

=.1660. Since this is a p-value, if takes a

typical value like .05 or .10, we can say p-value and not

reject the null hypothesis. )

7. Hypothesis Test for Means - Rare Events

In statistics, ‘rare events’ is a code word for the Poisson distribution. The

easiest way to approach Poisson results is to use a p-value. For example

if you wish to test against and you have a result

that says , the p-value is .

(For an example, see 252oneslx2) {poiss}

8. Hypothesis Tests for a Variance.

To test against (For an example, see

252oneslx2) {chiSq}

i. Test Ratio: or

for large samples

ii. Critical Value:

(Don't try this for large samples.)

iii. Confidence Interval: or

for large samples

Appendix: One-sided and Two Sided Tests.

Assume the following:

so that .

A 2-sided Test:

(i) Test ratio: We test this against

two values of t, and .

We reject if is above or below . In this case we

do not reject . {ttable}

If we use p-value: . On the t-table 0.678 is

between and . So is between

.25 and .30 and the p-value is between .50 and .60. Since the p-value is

above we do not reject .

(ii) Critical value for :

. We reject if is above the upper critical value

or below the lower critical value

. In this case and we do not reject .

(iii) Confidence interval: This interval is 11.28 to 12.72. Since

is between these two limits, we do not reject .

A Left -Sided test:

(i) Test ratio: (We use the same data

as the two-sided problem) We test this against one value of t,

. We reject if is below . In this

case we do not reject .

If we use p-value: . On the t-table 0.678 is between

and . So is between .25 and .30

and the p-value is between .25 and .30. Since the p-value is above

we do not reject .

(ii) Critical value for : We reject if is below In

this case and we do not reject .

(iii) Confidence interval. Since the alternate hypothesis is ,

use Since

does not contradict we do not reject .

A Right-sided Test:

(i) Test ratio: We test this against

one value of t, . We reject if is above .

In this case we do not reject .

If we use p-value: . On the t-table 0.678 is between

and . So is between .25 and .30,

is between .70 and .75 and the p-value is between .70 and

.75. Since the p-value is above we do not reject .

(ii) Critical value for : We reject if is above

In this case and we do not reject .

(iii) Confidence interval: Since the alternate hypothesis is

, use Since does not contradict

, we do not reject .