Hypothesis Testing For  With  Known

HYPOTHESIS TESTING Basic idea: You want to see whether or not your data supports a statement about a parameter of the population. –The statement might be: The average age of night students is greater than 25 (  >25) To do this you take a sample and compute

If < 25 –You did not prove  >25. If >>25 –You are probably satisfied that  >25. If is slightly > 25 –You are probably not convinced that  >25. (although there is some evidence to support this)

Being Convinced serving on a jury.Hypothesis testing is like serving on a jury. –The prosecution is presenting evidence (the data) to show that a defendant is guilty (a hypothesis is true). “reasonable doubt”. –But even though the evidence may indicate that the defendant might be guilty (the data may indicate that the hypothesis might be true), you must be convinced beyond a “reasonable doubt”. Otherwise you find the defendant “not guilty”– this does not mean he was innocent, (there is not enough evidence to support the hypothesis – this does not mean that the hypothesis is not true, just that there was not enough evidence to say it was true beyond a reasonable doubt).

When Can You Conclude  > 25? “a lot greater”You are convinced  >25 if you get an that is “a lot greater” than 25. “a lot”How much is “a lot” ? –This is hypothesis testing. above

“<” Tests Can you conclude that the average age of night students is less than 27? (  < 27) below

“  ” Tests Can we conclude that the average age of students is different from 26? (   26) away from

Hypothesis Testing Five Step Procedure 1.Define Opposing Hypotheses. (  ) 2.Choose a level of risk (  ) for making the mistake of concluding something is true when its not. 3.Set up test (Define Rejection Region). random sample 4.Take a random sample. 5.Calculate statistics and draw a conclusion.

STEP 1: Defining Hypotheses H 0 (null hypothesis -- status quo) H A (alternate hypothesis -- what you are trying to show) Can we conclude that the average age exceeds 25? H 0 :   25 H A :  > 25

Test H 0 “At the break point”  = 25 Test the hypothesis at the point that would give us the most problem in deciding if  >25. 15very unlikely false –If  really were 15, it would be very unlikely that we would draw a sample of students that would lead us to the false conclusion that  >25. 22more likely false 24 more likely24.9 –If  really were 22, it is more likely that we would draw a sample with a large enough sample mean to lead us to the false conclusion that  >25; if  really were 24, it would be even more likely, 24.9 even more likely. –  = 25 false –  = 25 is the “most likely” value in H 0 of generating a sample mean that would lead us to the false conclusion that  >25. So H 0 is tested “at the breakpoint”,  = 25

STEP 2: Choosing a Measure of Risk (Selecting  )  = P(concluding H A is true when it is not) Typical  values are.10,.05,.01, but  can be anything. –Values of  are often specified by professional organizations (e.g. audit sampling normally uses values of.05 or.10).

STEP 3: How to Set Up the Test Defining the “Rejection Region” Depends on whether H A is a >, <, or  hypothesis. The “> Case”  > 25. –Suppose we are hypothesizing  > 25. –When a random sample of size n is taken: If  really = 25, then there is only a probability =  that we would get an value that is more than z  standard errors above 25. greater thanThus if we get an value that is greater than z  standard errors above 25, we are willing to conclude  > 25. x crit –We call this critical value x crit.

Rejecting H 0 (Accepting H A )0Z 25  =25 if H 0 is true REJECTION REGION Reject H 0 (Accept H A ) if we get zzzz 

How many standard errors away is ? test statisticThis is called the test statistic, z

A ONE-TAILED “>” TEST Reject H 0 (Accept H A ) if: z > z  Values of z   z 

A ONE-TAILED “<“ TEST If H A were, H A :  < 27 The test would be: Reject H 0 (Accept H A ) if: z < -z  Values of -z   -z 

A TWO-TAILED “  ” TEST If H A were, H A :   26 The test would be: Reject H 0 (Accept H A ) if: z z  /2 Values of -z   -z  /2 z  /

STEP 4: TAKE SAMPLE randomAfter designing the test, we would take the sample according to a random sampling procedure.

STEP 5: CALCULATE STATISTISTICS From the sample we would calculate Then calculate:

DRAWING CONCLUSIONS One Tail Tests “>” TESTH A :  > 25“>” TEST: H A :  > 25 z > z  (Reject H 0 ) If z > z  -- Conclude H A is true (Reject H 0 ) z < z  (Do not reject H 0 ) If z < z  -- Cannot conclude H A is true (Do not reject H 0 ) “<” TEST:H A :  < 27“<” TEST: H A :  < 27 z < -z  If z < -z  -- Conclude H A is true (Reject H 0 ) z > -z  (Do not reject H 0 ) If z > -z  -- Cannot conclude H A is true (Do not reject H 0 )

DRAWING CONCLUSIONS Two Tail Tests H A :   26 Case 1 -- When z > 0z to z  /2Case 1 -- When z > 0: compare z to z  /2 z > z  /2 If z > z  /2 -- Conclude H A is true (Reject H 0 ) z < z  /2 (Do not reject H 0 ) If z < z  /2 -- Cannot conclude H A is true (Do not reject H 0 ) Case 2 -- When z < 0Case 2 -- When z < 0: compare z to -z  /2 z < -z  /2 If z < -z  /2 -- Conclude H A is true (Reject H 0 ) z > -z  /2 (Do not reject H 0 ) If z > -z  /2 -- Cannot conclude H A is true (Do not reject H 0 )

DRAWING CONCLUSIONS Two Tail Tests (Alternative) H A :   26 Compare |z| to z α/2Cases 1 and 2 can be combined by simply looking at |z|. The test becomes: Compare |z| to z α/2 |z| > z  /2 If |z| > z  /2 -- Conclude H A is true (Reject H 0 ) |z|< z  /2 (Do not reject H 0 ) If |z|< z  /2 -- Cannot conclude H A is true (Do not reject H 0 )

Examples Suppose –We know from long experience that  = 4.2 –We take a sample of n = 49 students –We are willing to take an  =.05 chance of concluding that H A is true when it is not (Note: z.05 = 1.645) Because our sample is large, a normal distribution approximates the distribution of

Example 1: Can we conclude  > 25? 1. H 0 :  = 25 H A :  >  =.05 3.Reject H 0 (Accept H A ) if: 4. Take sample 25,21,… 33.

Example 2: Can we conclude  < 27? 1. H 0 :  = 27 H A :  <  =.05 3.Reject H 0 (Accept H A ) if: 4. Take sample 22,28,… 33. NOTE!

Example 3: Can we conclude   26? 1. H 0 :  = 26 H A :   26 (This is a two-tail test) 2.  =.05 3.Reject H 0 (Accept H A ) if: 4. Take sample 25,21,… 33. NOTE!

> TESTS Given Values > Can conclude mu > 25 =NORMSINV(1-C2) =AVERAGE(A2:A50) (C7-C3)/(C1/SQRT(49))

< TESTS > Cannot conclude mu < 27 Given Values =-NORMSINV(1-C2) =AVERAGE(A2:A50) (C14-C10)/(C1/SQRT(49))

 TESTS Given Values < Cannot conclude mu  26 =ABS((C21-C17)/(C1/SQRT(49))) =NORMSINV(1-C2/2) =AVERAGE(A2:A50)

REVIEW “Common sense concept” of hypothesis testing 5 Step Approach –1. Define H 0 (the status quo), and H A (what you are trying to show.) –2. Choose  = Probability of concluding H A is true when its not. –3. Define the rejection region and how to calculate the test statistic. –4. Take a random sample. –5. Calculate the required statistics and draw conclusion. There is enough evidence to conclude H A is true (Reject H 0 ) There is not enough evidence to conclude H A is true (Do not reject H 0 ).