HYPOTHESIS TESTING



HYPOTHESIS TESTING

A hypothesis is a claim. Usually there are two rival claims.

Shower hypothesis: Water temperature is acceptable /not

How to Develop a Hypothesis?

We need to translate a problem into a statement involving a statistical measure. The measure or a parameter like ( or p is then used in the derivation of hypothesis. For example, the CLAIM may be that education increases earnings so ( =average earnings of the educated. If the average earnings of the entire population is known to be $30,000

Now the two rival claims are:

H0 ( ( 30000 versus Ha ( > 30000 .

H0 is called the null hypothesis. It says that the educated earn no better than others.

Ha is called the alternative hypothesis, which says that the educated earn more.

In this example, the alternative is one-sided. If the claim were that the educated earn LESS than others, then the alternative would be set up as Ha ( < 30000.

Hypothesis testing is a scientific method of choosing between two claims H0 and Ha

Some principles:

1) H0 is presumed true unless overwhelming evidence rejects it. E.g., H0: defendant is not guilty!

2)Sample data give test statistics for (, p or (2

3)We reject the Null H0 if statistic falls in Rejection Region

4) Null is usually a zero value (hence the name null).

5) Instead of saying ACCEPT one says FAILS TO REJECT.

6) Absolute certainty does not exist.

If there is a standard value for ( the null is H0 ( =std value (or true value). e.g. H0 ( =10 and two-sided alternative is Ha ( ( 10, where it could be larger than 10 or smaller than 10.

Need Skill to decide (1) appropriate statistical parameter (, p, etc. (2) appropriate Null (3) One-sided or two-sided.

When one of the alternatives is selected, there can be error.

Type I (() and Type II (() errors

Definition: Type I error: Selecting Ha when H0 should be selected.

H0 DOGS are dead, Ha DOGS are alive

Truth is DOGS are dead, still selecting Ha is Type I error

Definition: Type II error: Selecting H0 when Ha should be selected.

H0 DOGS are dead, Ha DOGS are alive

Truth is DOGS are alive, still selecting H0 means Type II error.

Definition: ( denotes the probability of Type I error = This is also called the “Level of test.”

Definition: ( denotes the probability of Type II error. This is usually hard to determine since it depends on unknown parameter ( itself.

It is desirable to formulate the hypothesis so that ( is the most serious consequence. The two types of errors are inversely related. There is a trade-off between ( and (. The smaller we make ( the larger the ( we have to accept. Hence one usually chooses ( largest tolerable !

Steps in the Test of Hypothesis

1.Define the hyp. to be tested in plain English.

2.Select the appropriate statistical measure (such as ( p, (2) to rephrase the hypothesis.

3.Determine whether hyp. should be 1 or 2-sided.

4.State the hypothesis using the statistical measure selected in step 2.

5.Specify (, the “level” of the test.

6.Select the appropriate test statistic, based on the information at hand and the assumptions you are willing to make.

7.Determine the critical value of the test statistic. Three factors for critical value:

a. the type of alternative hypothesis,

(1) Two-sided (2) 1-sided left (3) 1-sided Right

b. the specification of a, the level of the test,

c. the distribution of the test statistic.

8.Collect sample data and compute the value of the test statistic.

9.Make the decision. Is the value of the test statistic in the rejection region?

a. If yes, reject the null hypothesis in favor of the alternative.

b.If no, do not reject the null hypothesis.

10. State conclusion in terms of the orig. question.

11-5 Testing a Hypothesis about a Population Mean (

Example illustrates the hypothesis testing procedure.

A local school board member wants to know if sophomore students at Lincoln High School have approximately the same reading level as the state average for tenth graders. The state average is150 words per minute with a standard deviation of 15. The level of the test is to be set at .05. A random sample size of 100 tenth graders has been drawn, and the resulting average is 157 words per minute.

Step 1 (Define the hypothesis in plain English.)

The hypothesis is straight forward:

• Lincoln High School tenth graders are reading at the state average.

• They are not.

Step 2 (select statistical measure) ( =average of something. Number of words read per minute by Lincoln Hi sophomores.

Step 3 (one or 2 sided) They want to know either way, above or below average, so 2-sided.

Step 4 ( State H0 Ha ) H0 ( =150 the std. value and Ha ( ( 150

Step 5 (Level of test?) The problem statement says ( = 0.05 or Type I error of 5%

Step 6 (select statistic) The choice depends on answer to 3 questions. (i) Is ( known? (ii) Is the distribution Normal? (iii) is n large?

Case 1: If ( is known, n is large, and distribution is Normal, use Z=([pic]( ( 0 ) / ( ,

where ( =( / ( n

Case 2: If ( is known, n is large, but the distribution may not be Normal. Still, since n>30 Central Limit Theorem suggests same Z statistic.

Case 3: If ( is unknown, Normality not assumed and n>30 we can simply use s instead of ( in the above formula.

Step 7 (Determine critical value) For 5%, level z=1.96. The rejection region is in Tails. “Fail to Reject” region is at the center. Why?

Recall H0 ( =150 is the null. If true, the difference ([pic] ( () should be near zero.

Since [pic] is likely to be normal, its z-transform is N(0,1), or the standard normal variable. Hence from Normal Tables we can decide that if [pic] > 1.96 or [pic] < (1.96 it belongs in Tails where we decide to reject the null of zero.

Step 8 (Compute the test statistic) If sample mean [pic] =157 we have Z= (157 ( 150)/ SE, where SE=( / ( n = 15/10=1.5, Now Z=4.67. This means that [pic] is 4.67 standard deviations away from hypothesized value of zero. This seems too far away from the value of the null. Is it too far from formal statistical viewpoint?

Step 9 (Make a decision) Critical value is 1.96 the test statistic is much larger than critical value 1.96 Decision must be to reject the null H0. The difference between [pic] and ( could not have been caused by mere sampling variation. We decide that the observed difference is statistically significant. That is, the difference is rather rare for a 5% test, so unlikely.

Step 10: (State conclusion) There is significant evidence at the 5% level that tenth graders at Lincoln Hi do not read at the state average. It would be tempting to conclude that they read at a higher level, but we did not set up our test as a one-sided test. The conclusions must be consistent with the hypothesis.

P-Values

Definition: The probability of observing a value as extreme or more extreme than the observed value.

It is computed as the tail area with reference to the observed test statistic. For one-sided test, it is the left or the right tail. If left tail, and test statistic is z1=(-1) then p value is given by the R command

> pnorm(-1)

[1] 0.1586553

If right tail and z1=1.5, say then p value is

given by the R command:

> pnorm(1.5, lower.tail=FALSE)

[1] 0.0668072

If population variance (or standard deviation) is unknown and therefore the variance (or sd) is estimated from the sample, then Student's t distribution should be used instead of Normal density.

For example, if the t statistic is t1=(-1.21) one-sided left tail with degrees of freedom 20 then the p value is obtained by the following R command:

> pt(-1.21, df=20)

[1] 0.1201933

If the right side test with t1=2.1, df=33 then use following R command

> pt(2.1, df=33, lower.tail=FALSE)

[1] 0.02172731

For two-sided test P-value is computed as the sum of the two tail areas or by simply doubling the above computations since z and t distributions are symmetric.

Decisions: If ( > P-value Reject the Null. For example, for a 95% test with (=0.05 we reject the null if the P-value tail is small, say 0.04. If the tail is small, the observed value is obviously in the rejection region.

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