p-Values forp-Values for

Hypothesis Testing About Hypothesis Testing About With With Known Known

Hypothesis Testing(Revisited)

• Five Step Procedure1. 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).

4. Take a random samplerandom sample.

5. Calculate statistics and draw a conclusion.

Concept of a p-value• Ignore step 3• Calculate x

The probability of getting an value

at least as far away as the observed

value, if Hif H00 were true were true.

p-Value

xx

Calculating p-Values

• A p-value is a probability whose definition varies depending on the type of test we are doing (i.e. the form of the alternate hypothesis.)

Alternate Hypothesis p-value

“>” Area to the right of

“<” Area to the left of

“” 2* Area in the “tail”

(to the right or left of )x

xx

6.49

4.2σX

25 25

X

p-value For “> Tests” = P(Getting a value greater than When H0 is true)

H0: = 25

HA: > 25

x

23.26x

23.26xget and

sample random a Take

p-value =26.23)XP(

= 4.2, n = 49

0 Zz = (26.23-25)/.6

2.05

.9798p = 1-.9798

= .0202.0202

6.49

4.2σX

27 27

X

p-value For “< Tests” = P(Getting a value less than When H0 is true)

H0: = 27

HA: < 27

x

23.26x

23.26xget and

sample random a Take

p-value =26.23)XP(

= 4.2, n = 49

0 Zz = (26.23-27)/.6

-1.28

p = .1003.1003

6.49

4.2σX

X26 26

p-value For “ Tests” = P(Getting a value at least as far away as When H0 is true) H0: = 26

HA: 26

x

= 4.2, n = 49

0 Z

.38

23.26xget and

sample random a Take

23.26x .23 below 26 .23 above 26

77.25

z = (26.23-26)/.6

p-value =Area above 26.23 +Area below 25.77 =2*Area above 26.23

.6480.3520 .3520

p = 2(.3520)

= .7040.7040

P-VALUES AND α

• Consider HA: > 25– Here we got z = 2.05 – Since = .05, z.05 = 1.645 Can Accept HA

– Suppose = .01; z.01 = 2.326 Cannot Accept HA

– What about = .02? z.02 = 2.054 Cannot Accept HA

– What about = .03? z.03 = 1.88 Can Accept HA

• There is some value of that is the “break-point” between accepting and not accepting HA-- this is the p-valuep-value. If p α, Accept HAccept HAA

If p > α, Do Not Accept HDo Not Accept HAA

LOW p-values are SIGNIFICANT!!

=AVERAGE(A2:A50)

=(C6-C3)/(C2/SQRT(49))

=1 – NORMSDIST(C7)

=AVERAGE(A2:A50)

=(C6-C3)/(C2/SQRT(49))

=NORMSDIST(C7)

=AVERAGE(A2:A50)

=(C6-C3)/(C2/SQRT(49))

=2*(1-NORMSDIST(C7))

Note: If z were negative, the p-value would have been:

=2*NORMSDIST(C7)

REVIEW

• p-values measure the strength of the test– lower p-values indicate more strongly that HA is true

• p-values– “>” tests -- Area in upper tail (to the right of ) – “<” tests -- Area in lower tail (to the left of )– “” tests -- twice the area in a “tail”

• If z >0 -- twice the area in the upper tail

• If z< 0 -- twice the area in the lower tail

x

x