Hypothesis Testing P-Value Method

Hypothesis Testing MAT-150: Statistics 1 Spring2015 Page: 1
(P-Value Method)
Bergen Community College Cerullo Learning Assistance Center (CLAC) 201-879-7489
Definition
TextbookDefinition:The P-value (orprobabilityvalue) isthe probabilityof gettingasample statistic
(suchas the mean) or a more extreme samplestatisticinthe directionof the alternative hypothesis
whenthe null hypothesisistrue.
The difference betweenthe traditional methodand the p-value method:
 The traditional methodcomparesthe test value/statistictothe critical value
 The p-value method comparesthe p-value (aprobability)tothe significancelevel (𝛼value)
Determine which Test
z-Testfor Mean
Whensigma(σ),a.k.athe population standarddeviation,isknown(given) andthe problemaskstotesta
claimdealingwithamean
z-Testfor Proportion
Whenthe problemasksto testa claimdealingwithaproportion;if the problemistalkingabout
percentages,ormentions some type of proportion (i.e. 84studentsoutof a randomsample of 100 BCC
studentspassedtheirmathcourse),thistestwillmostlikelybe used
t-Testfor Mean
Whensigma(σ),a.k.athe populationstandarddeviation,isunknown(notgiven),butinsteadthe sample
standarddeviation(s) isknown(given)andthe problemaskstotesta claimdealingwithamean
Χ2
-Testfor Variance or Standard Deviation
Whenthe problemasksto testa claimdealingwithapopulationvariance (σ2
) orapopulationstandard
deviation(σ) [recall:variance =(standarddeviation)2
]

(P-Value Method)
Gather information from problem & check assumptions
z-Testfor Mean
µ = populationmean
𝑥̅ = sample mean [recall: if given a bunch of scores, 𝑥̅ =
Σ ( 𝑥) [sum of all scores]
𝑛 [sample size]
]
σ = populationstandarddeviation
n = sample size
𝛼 = level of significance
Assumptions
1. The sample isa randomsample
2. Eithern ≥ 30 or the populationisnormallydistributedif n < 30
z-Testfor Proportion
p = populationproportion
q = 1 – p
𝑝̂ = sample proportion [if not directly given, 𝑝̂ =
𝑥 (number of successes)
𝑛 (sample size)
; i.e. 84 students 100 BCC students, 𝑝̂ =
84
100
]
n = sample size
Assumptions
2. The conditionsforabinomial experimentare satisfied [page437 8th
edition:explanation;referto
chapter5 section3 page 271)
3. np ≥ 5 and nq ≥ 5

(P-Value Method)
t-Testfor Mean
µ = populationmean
𝑥̅ = sample mean [recall: if given a bunch of scores, 𝑥̅ =
Σ ( 𝑥) [sum of all scores]
𝑛 [sample size]
]
s = sample standarddeviation [recall: if given a bunch of scores, 𝑠 = √
𝑛(Σ 𝑥2) −(Σ 𝑥)2
𝑛 (𝑛−1)
]
n = sample size
n-1 = degreesof freedom
Assumptions
2. Eithern ≥ 30 or the populationisnormallydistributedif n < 30
Χ2
σ2
= populationvariance [remember:populationvariance =(populationstandarddeviation)2
s2
= sample variance [remember:samplevariance =(sample standarddeviation)2
[recall: if given a bunch of scores, 𝑠2 =
𝑛(Σ 𝑥2) −(Σ 𝑥)2
𝑛 (𝑛−1)
]
n = sample size
n-1 = degreesof freedom
Assumptions
1. The sample mustbe randomly selectedfromthe population
2. The populationmustbe normallydistributedforthe variable understudy
3. The observationsmustbe independentof one another

(P-Value Method)
Procedure for each type of Test (P-value method)
Stepsfor z-Test(Mean)
1. State the hypotheses [ null hypothesis(Ho) &alternative hypothesis(H1 orHa) ] and identifythe
claim:
 Ho: µ ("=") some populationmean
i. Note:dependingonprofessor,the symbol usedforthe null hypothesiswill
alwaysbe "=" OR the symbol couldeitherbe one of the followingoptions
"=" ; "≥" ; "≤"
1. If professorprefersusingone of the three symbols("=";"≥" ; "≤"), in
orderto determine which one touse,referto the alternativehypothesis
and use the opposite symbol
a. The opposite symbolsfor"≠" is"="
i. Example: Ho: µ = 100
Ha:µ ≠ 100
b. The opposite symbolsfor">"is"≤"
i. Example: Ho: µ ≤ 100
Ha:µ > 100
c. The opposite symbolsfor"<"is"≥"
i. Example: Ho: µ ≥ 100
Ha:µ < 100
ii. Note:the null hypothesis will alwayscontainsome type of equalitysymbol
1. Refertothe table below forthe differentcommonphrasesfor
"=" ; "≥" ; "≤"
 Ha:µ ("≠" ; ">" ; "<") some populationmean
i. Note:in orderto determine whichsymbol touse lookoutforcommonphrases
(refertotable below)
ii. Note:if ">" (right-tailed) or"<"(left-tailed) isused,the testisaone-tailedtest;
if "≠" is used,the testbecomesatwo-tailedtest
Common Phrases used for Hypothesis Testing
> < = ≠
is greater than is less than is equal to is not equal to
is above is below is the same as is different from
is higher than is lower than has not changed from has changed from
is longer than is shorter than is the same is not the same
is increased is smaller than is “is not”
is too low Is decreased or reduced is equivalent to
Is too high
≥ ≤
no less than no more than
at least at most

(P-Value Method)
2. Compute the test value/statistic (a.k.athe z-score):
 Write downall informationgiveninproblem(referto"gatherinformationsection")
 Plug information intoformula:z=
𝑥̅− 𝜇
𝜎
√ 𝑛
OR
(𝑥̅− 𝜇)√ 𝑛
𝜎
3. Findthe p-value:
 Easiestwayto findthe p-value forz-score (yourteststatistic),whichwasfoundinthe
previousstep:
i. Referto the z-table (Table E)
ii. Look up the negative value (if notalreadynegative)of the z-score (test
statistic) onthe z-table (Table E) inorderto findthe corresponding proportion
iii. Thisproportion(thatwasfoundinside Table E) isyourp-value
1. Note:if the testis a two-tailedtest,thenyouhave to double the p-value
4. Make the decisionto either"reject"or"donot reject"the null hypothesis(Ho):
 If p-value ≤ 𝛼 : rejectHo
 If p-value > 𝛼 : do not rejectHo
5. Summarize Results:
 There are fourdifferentpossibilitiesforthe conclusion:
i. If the claim isin Ho and Ho isrejected:There is enough/sufficientevidence to
rejectthe claimthat…
ii. If the claim isin Ho and Ho isnot rejected: There is not enough/insufficient
evidence torejectthe claimthat…
iii. If the claim isin H1 and Ho is rejected:There is enough/sufficientevidence to
support the claimthat …
iv. If the claim isin H1 and Ho is not rejected:There is not enough/insufficient
evidence to supportthe claimthat …

(P-Value Method)
Stepsfor z-Test(Proportion)
claim:
a. Ho: p ("=") some populationmean
"=" ; "≥" ; "≤"
i. Example: Ho: p = 100
Ha:p ≠ 100
i. Example: Ho: p ≤ 100
Ha:p > 100
i. Example: Ho: p ≥ 100
Ha:p < 100
"=" ; "≥" ; "≤"
b. Ha:p ("≠" ; ">" ; "<") some population proportion
> < = ≠
Is too high
≥ ≤
at least at most

(P-Value Method)
2. Compute the test value/statistic (a.k.athe z-score):
a. Write downall informationgiveninproblem(referto"gatherinformationsection")
b. Plug information intoformula:z=
𝑝̂ − 𝑝
√
𝑝 ∙ 𝑞
𝑛
OR
( 𝑝̂ − 𝑝)√ 𝑛
√ 𝑝 ∙ 𝑞
3. Findthe p-value:
a. Easiestwayto findthe p-value forz-score (yourteststatistic),whichwasfoundinthe
previousstep:
i. Referto the z-table (Table E)
ii. Look up the negative value (if notalreadynegative)of the z-score (test
statistic) onthe z-table (Table E) inorderto findthe correspondingproportion
iii. Thisproportion(thatwasfoundinside Table E) isyourp-value
1. Note:if the testis a two-tailedtest,thenyouhave to double the p-value
a. If p-value ≤ 𝛼 : rejectHo
b. If p-value > 𝛼 : do not rejectHo
a. There are fourdifferentpossibilitiesforthe conclusion:
***NOTICE: the differences between z-Test (Mean) and z-Test (Proportion): "p" is used in the
hypothesesand the formulausedis z=
𝑝̂ − 𝑝
√
𝑝 ∙ 𝑞
𝑛
(so,make sure youwrite downthe correct information)***

(P-Value Method)
t-Testfor Mean
claim:
 Ho: µ ("="OR "=" ; "≥" ; "≤") some populationmean
"=" ; "≥" ; "≤"
1. If professorprefersusingone of the three symbols ("=";"≥" ; "≤"), in
i. Example: Ho: µ = 100
Ha:µ ≠ 100
i. Example: Ho: µ ≤ 100
Ha:µ > 100
i. Example: Ho: µ ≥ 100
Ha:µ < 100
"=" ; "≥" ; "≤"
 Ha:µ ("≠" ; ">" ; "<") some populationmean
> < = ≠
Is too high
≥ ≤
at least at most

(P-Value Method)
2. Compute the test value/statistic (a.k.athe t-score):
 Plug information intoformula:t=
𝑥̅− 𝜇
𝑠
√ 𝑛
OR
(𝑥̅− 𝜇)√ 𝑛
𝑠
3. Findthe p-value:
 t-Testsinvolve p-valueintervals;inordertofind thisinterval fora t-score (yourtest
statistic),whichwasfoundin the previousstep:
i. Referto the t-table (Table F)
ii. Findthe row that correspondsto the problem’s degreesoffreedom(d.f =n-1)
iii. Findthe two valuesthat your t-score (teststatistic) fallsbetween
iv. Look upto the row labeledone-tail ortwo-tailtofind 𝛼 valuesforyourp-value
interval (refertorow that correspondsto the problem)
1. Recall: If ">" or "<" isused,the testisa one-tailedtest
If "≠" is used,the testbecomesatwo-tailedtest
v. Create yourp-value interval withthe two 𝛼 valuesfound:
(lower bound/number) < p-value < (upper bound/number)
vi. For example (refertopicture below forvisual help):Findthe p-valuewhenthe
teststatistic(t-score) is2.056, the sample size is11, andthe testisright-tailed
1. RefertoTable F
2. Next,goto row withd.f = 10 (n-1 = 11 - 1)
3. Findtwovaluesthat2.056 fallsbetween:itfallsbetween1.812 and
2.228
4. Then,lookupto the row labeledone-tail since the problemisaright-
tailedtest
a. The two valuesare 0.05 and 0.025 whichcreate yourp-value
interval: 0.025 < p-value < 0.05

(P-Value Method)
 If upper bound of p-value ≤ 𝛼: rejectHo
 If lower bound of p-value > 𝛼: do not rejectHo
i. For example:If p-value intervalis 0.025 < p-value <0.05 and 𝛼 = .1
1. We wouldrejectHo because 0.05 (upperbound) ≤ .1
ii. For example:If p-value intervalis 0.025 < p-value <0.05 and 𝛼 = 0.005
1. We donot rejectHo because 0.025 (lowerbound) > 0.005
 There are fourdifferentpossibilitiesforthe conclusion:

(P-Value Method)
Χ2
claim:
 Ho: 𝜎2 ("="OR "=" ; "≥" ; "≤") some population variance
OR Ho: 𝜎 ("=" OR "=" ; "≥" ; "≤") some population standarddeviation
"=" ; "≥" ; "≤"
i. Example: Ho: 𝜎2 = 100 OR Ho: 𝜎 = 100
Ha: 𝜎2 ≠ 100 OR Ha: 𝜎 ≠ 100
i. Example: Ho: 𝜎2 ≤ 100 OR Ho: 𝜎 ≤ 100
Ha: 𝜎2 > 100 OR Ha: 𝜎 > 100
i. Example: Ho: 𝜎2 ≥ 100 OR Ho: 𝜎 ≥ 100
Ha: 𝜎2 < 100 OR Ha: 𝜎 < 100
"=" ; "≥" ; "≤"
 Ha: 𝜎2 ("≠" ; ">" ; "<") some population variance
OR Ha: 𝜎 ("≠" ; ">" ; "<") some population standard
if "≠" is used,the testbecomesa two-tailedtest
> < = ≠
Is too high
≥ ≤
at least at most

(P-Value Method)
2. Compute the test value/statistic (a.k.athe Χ2
-score):
 Plug information intoformula:Χ2
=
( 𝑛−1) 𝑠2
𝜎2
3. Findthe p-value:
 Χ2
-Testsinvolve p-value intervals;inordertofind thisinterval foraΧ2
-score (yourtest
statistic),whichwasfoundinpreviousstep:
i. Referto the Χ2
-table (Table G)
ii. Findthe row that correspondsto the problem’s degreesoffreedom(d.f =n-1)
iii. Findthe two valuesthat your Χ2
-score (teststatistic) fallsbetween
iv. Look upto the toprow to find 𝛼 valuesforyourp-value interval
1. If right-tailedtest (𝛼valuesontop row from 0.10 to 0.005), these 𝛼
valueswill create p-value interval
a. Recall:">" meansright-tailedtest
2. If left-tailedtest(𝛼valuesontop row from 0.995 to 0.90), youneedto
subtract each 𝛼 value from1; those values will create p-valueinterval
a. Recall:"<" meansleft-tailedtest
3. If two-tailedtest,each 𝛼 value needs tobe doubled;those valueswill
create p-interval
a. Recall:"≠" meanstwo-tailedtest
b. Note:If teststatistic(Χ2
-score) fallsonthe rightside of the
distribution(toprow valuesfrom 0.10to 0.005), double each 𝛼
value
c. Note:If teststatistic(Χ2
-score) fallsonthe leftside of the
distribution(toprow valuesfrom 0.995 to 0.90), firstsubtract
each 𝛼 value from1, thendouble it
v. Create yourp-value interval withthe two 𝛼 valuesfound:
(lower bound/number) < p-value < (upper bound/number)
vi. Examples foreachsituationare on the nextcouple pages
1. NOTICE: p-value intervalsforΧ2
-Testswill usuallyhave valuesbetween
0.005 and 0.10 if it's a one-tailedtest;if it'satwo-tailedtest,the values
will be between0.01 and0.20
a. Note:This isvalidforTable G that's usedin Bluman'stextbook
2. TRICK: make the toprow of Table G (Χ2
Table),startingfromleftside,
0.005, 0.01, 0.025, 0.05, 0.10, 0.10, 0.05, 0.025, 0.01, 0.005 (referto
picture below)
a. If you choose to use thistrick,youwill not have to worry about
subtractingfrom"1"
i. However,if two-tailedyoustill needtodouble the
valuesbefore creatingyourp-value interval

(P-Value Method)
1. Example involvingright tail (refertopicture below forvisual help):Findthe p-value whenthe
teststatistic(Χ2
-score) is 19.274, the sample size is8, and the testis right-tailed
a. First, refertoTable G
b. Next,goto row withd.f = 7 (n-1= 8 - 1)
c. Findtwovaluesthat 19.274 fallsbetween:itfallsbetween 18.475 and 20.278
d. Then,lookupto the toprow and since it’sa right-tailedtest,thesevalues willconstruct
your p-value interval
i. The two values are 0.01 and 0.005 sothe p-value interval is
0.005 < p-value < 0.01
2. Example involvinglefttail (refertopicture below forvisual help):Findthe p-value whenthe
teststatistic(Χ2
-score) is2.940, the sample size is11, and the testis left-tailed
a. First,refertoTable G
b. Next,goto row withd.f = 10 (n-1 = 11 - 1)
c. Findtwovaluesthat2.940 fallsbetween:itfallsbetween2.558 and 3.247
d. Then,lookupto the toprow and since it’sa left-tailedtest,thesevalueswillneedtobe
subtractedfrom1 inorderto construct your p-value interval
i. The two valuesare 0.99 and 0.975, so subtract these valuesfrom 1:
1. 1 - 0.99 = 0.01
2. 1 - 0.975 = 0.025
ii. The twovaluesusedforthe p-value intervalare 0.01 and 0.025
1. So, the p-value interval is 0.01 < p-value < 0.025

(P-Value Method)
3. Example involvingtwo-tailson leftside of the distribution(refertopicture below forvisual
help):Findthe p-valuewhenthe teststatistic(Χ2
-score) is 0.521, the sample size is8, andthe
testis two-tailed
c. Findtwovaluesthat 0.521 fallsbetween:itfallsbetween 0.412 and 0.554
d. Then,lookupto the toprow and since it’sa two-tailedtestonthe leftside of the
distribution,these valueswill needtobe subtractedfrom1, thendoubled inorderto
construct yourp-value interval
i. The two valuesare 0.995 and0.99, so firstsubtract these valuesfrom 1,then
double the values:
1. 1 - 0.995 = 0.005 then 2 (0.005) = 0.01
2. 1 - 0.99 = 0.01 then 2 (0.01) = 0.02
ii. The two valuesusedforthe p-value interval are 0.01 and 0.02
1. So, the p-value interval is0.01 < p-value < 0.02
4. Example involvingtwo-tailson right side of the distribution(refertopicture below forvisual
help):Findthe p-valuewhenthe teststatistic(Χ2
-score) is8.420, the sample size is5, andthe
testis two-tailed
c. Findtwovaluesthat8.420 fallsbetween:itfallsbetween 7.779 and 9.488
d. Then,lookupto the toprow and since it’sa two-tailedtestonthe rightside of the
distribution,these valueswill needtobe doubledinordertoconstructyour p-value
interval
i. The two valuesare 0.10 and 0.05, so double these values:
1. 2 (0.10) = 0.20
2. 2 (0.05) = 0.10
ii. The two valuesusedforthe p-value interval are 0.20 and 0.10
1. So,the p-value interval is0.10 < p-value < 0.20

(P-Value Method)
a. If upper bound of p-value ≤ 𝛼: rejectHo
b. If lower bound of p-value > 𝛼: do not rejectHo
i. For example:If p-value interval is .005 < p-value < .01 and 𝛼 = .1
1. We wouldrejectHo because .01 (upperbound) ≤ .1
ii. For example:If p-value interval .005< p-value <.01 and 𝛼 = .001
1. We donot rejectHo because .005 (lowerbound) > .001
a. There are fourdifferentpossibilitiesfor the conclusion:
***NOTICE: Test values/statisticscaneitherbe az-score, t-score, or Χ2
-score; it depends on the type of
problem***
Bibliography
Bluman,AllanG."HypothesisTesting."ElementaryStatistics:A StepbyStepApproach,SeventhEdition.
7th ed.NewYork:McGrawhill HigherEducation,2009. N. pag.Print.

Hypothesis Testing P-Value Method

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