p-values for hypothesis testing about with known
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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
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