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Transcript of 1 1 Slide Chapter 9 Hypothesis Tests Developing Null and Alternative Hypotheses Developing Null and...
1 1 Slide Slide
Chapter 9 Hypothesis TestsChapter 9 Hypothesis Tests Developing Null and Alternative HypothesesDeveloping Null and Alternative Hypotheses Type I and Type II ErrorsType I and Type II Errors Population Mean: Population Mean: σσ Known Known Population Mean: Population Mean: σσ Unknown Unknown Population ProportionPopulation Proportion
2 2 Slide Slide
9.1 Developing Null and Alternative Hypotheses9.1 Developing Null and Alternative Hypotheses
Hypothesis testingHypothesis testing can be used to determine whether can be used to determine whether a statement about the value of a population parametera statement about the value of a population parameter should or should not be rejected.should or should not be rejected. The The null hypothesisnull hypothesis, , denoted by denoted by HH0 0 , , is a tentativeis a tentative assumption about a population parameter.assumption about a population parameter. The The alternative hypothesisalternative hypothesis, denoted by , denoted by HHaa, is the, is the opposite of what is stated in the null hypothesis.opposite of what is stated in the null hypothesis. The alternative hypothesis is what the test isThe alternative hypothesis is what the test is attempting to establish.attempting to establish.
3 3 Slide Slide
Testing Research HypothesesTesting Research Hypotheses
Developing Null and Alternative HypothesesDeveloping Null and Alternative Hypotheses
• The research hypothesis should be expressed asThe research hypothesis should be expressed as the alternative hypothesis.the alternative hypothesis.• The conclusion that the research hypothesis is trueThe conclusion that the research hypothesis is true comes from sample data that contradict the nullcomes from sample data that contradict the null hypothesis.hypothesis.
4 4 Slide Slide
Developing Null and Alternative HypothesesDeveloping Null and Alternative Hypotheses
Testing the Validity of a ClaimTesting the Validity of a Claim• Manufacturers’ claims are usually given the benefitManufacturers’ claims are usually given the benefit of the doubt and stated as the null hypothesis.of the doubt and stated as the null hypothesis.• The conclusion that the claim is false comes fromThe conclusion that the claim is false comes from sample data that contradict the null hypothesis.sample data that contradict the null hypothesis.
5 5 Slide Slide
Testing in Decision-Making SituationsTesting in Decision-Making Situations
Developing Null and Alternative HypothesesDeveloping Null and Alternative Hypotheses
• A decision maker might have to choose betweenA decision maker might have to choose between two courses of action, one associated with the nulltwo courses of action, one associated with the null hypothesis and another associated with thehypothesis and another associated with the alternative hypothesis.alternative hypothesis.
• Example: Accepting a shipment of goods from aExample: Accepting a shipment of goods from a supplier or returning the shipment of goods to thesupplier or returning the shipment of goods to the suppliersupplier
6 6 Slide Slide
One-tailedOne-tailed(lower-tail(lower-tail))
One-tailedOne-tailed(upper-tail)(upper-tail)
Two-tailedTwo-tailed
0 0: H 0 0: H
0: aH 0: aH 0 0: H 0 0: H
0: aH 0: aH 0 0: H 0 0: H
0: aH 0: aH
Summary of Forms for Null and Alternative Summary of Forms for Null and Alternative Hypotheses about a Population MeanHypotheses about a Population Mean
The equality part of the hypotheses always appearsThe equality part of the hypotheses always appears in the null hypothesis.in the null hypothesis. In general, a hypothesis test about the value of aIn general, a hypothesis test about the value of a population meanpopulation mean must take one of the followingmust take one of the following three forms (wherethree forms (where 00 is the hypothesized value ofis the hypothesized value of the population mean).the population mean).
7 7 Slide Slide
Example: Metro EMSExample: Metro EMS
Null and Alternative HypothesesNull and Alternative Hypotheses
Operating in a multipleOperating in a multiplehospital system with hospital system with approximately 20 mobile medicalapproximately 20 mobile medicalunits, the service goal is to respond to medicalunits, the service goal is to respond to medicalemergencies with a mean time of 12 minutes or less.emergencies with a mean time of 12 minutes or less.
A major west coast city providesA major west coast city providesone of the most comprehensiveone of the most comprehensiveemergency medical services inemergency medical services inthe world.the world.
8 8 Slide Slide
The director of medical servicesThe director of medical serviceswants to formulate a hypothesiswants to formulate a hypothesistest that could use a sample oftest that could use a sample ofemergency response times toemergency response times todetermine whether or not thedetermine whether or not theservice goal of 12 minutes or lessservice goal of 12 minutes or lessis being achieved.is being achieved.
Example: Metro EMSExample: Metro EMS
Null and Alternative HypothesesNull and Alternative Hypotheses
9 9 Slide Slide
Null and Alternative HypothesesNull and Alternative Hypotheses
The emergency service is meetingThe emergency service is meetingthe response goal; no follow-upthe response goal; no follow-upaction is necessary.action is necessary.
The emergency service is notThe emergency service is notmeeting the response goal;meeting the response goal;appropriate follow-up action isappropriate follow-up action isnecessary.necessary.
HH00::
HHaa::
where:where: = mean response time for the population= mean response time for the population of medical emergency requestsof medical emergency requests
10 10 Slide Slide
9.2 Type I Error9.2 Type I Error
Because hypothesis tests are based on sample data,Because hypothesis tests are based on sample data, we must allow for the possibility of errors.we must allow for the possibility of errors. A A Type I errorType I error is rejecting is rejecting HH00 when it is true. when it is true.
The probability of making a Type I error when theThe probability of making a Type I error when the null hypothesis is true as an equality is called thenull hypothesis is true as an equality is called the level of significancelevel of significance..
Applications of hypothesis testing that only controlApplications of hypothesis testing that only control the Type I error are often called the Type I error are often called significance testssignificance tests..
11 11 Slide Slide
Type II ErrorType II Error
A A Type II errorType II error is accepting is accepting HH00 when it is false. when it is false.
It is difficult to control for the probability of makingIt is difficult to control for the probability of making a Type II error.a Type II error. Statisticians avoid the risk of making a Type IIStatisticians avoid the risk of making a Type II error by using “do not reject error by using “do not reject HH00” and not “accept ” and not “accept HH00”.”.
12 12 Slide Slide
Type I and Type II ErrorsType I and Type II Errors
CorrectCorrectDecisionDecision Type II ErrorType II Error
CorrectCorrectDecisionDecisionType I ErrorType I Error
RejectReject HH00
(Conclude(Conclude > 12)> 12)
AcceptAccept HH00
(Conclude(Conclude << 12) 12)
HH0 0 TrueTrue(( << 12) 12)
HH0 0 FalseFalse(( > 12)> 12)ConclusionConclusion
Population Condition Population Condition
13 13 Slide Slide
9.3 9.3 pp-Value Approach to-Value Approach toOne-Tailed Hypothesis TestingOne-Tailed Hypothesis Testing
Reject Reject HH00 if the if the pp-value -value << ..
The The pp-value-value is the probability, computed using the is the probability, computed using the test statistic, that measures the support (or lack oftest statistic, that measures the support (or lack of support) provided by the sample for the nullsupport) provided by the sample for the null hypothesis.hypothesis. If the If the pp-value is less than or equal to the level of-value is less than or equal to the level of significancesignificance , the value of the test statistic is in the, the value of the test statistic is in the rejection region.rejection region.
14 14 Slide Slide
pp-Value Approach-Value Approach pp-Value Approach-Value Approach
p-valuep-value
00 -za = -1.28 -za = -1.28
= .10 = .10
zz
z =-1.46 z =-1.46
Lower-Tailed Test About a Population Mean:Lower-Tailed Test About a Population Mean: KnownKnown
Samplingdistribution of
Samplingdistribution of z x
n
0
/z x
n
0
/
pp-Value -Value << ,,so reject so reject HH00..
15 15 Slide Slide
pp-Value Approach-Value Approach pp-Value Approach-Value Approach
p-Valuep-Value
00 za = 1.75 za = 1.75
= .04 = .04
zz
z =2.29 z =2.29
Upper-Tailed Test About a Population Mean:Upper-Tailed Test About a Population Mean:σσKnownKnown
Samplingdistribution of
Samplingdistribution of z x
n
0
/z x
n
0
/
pp-Value -Value << ,,so reject so reject HH00..
16 16 Slide Slide
Critical Value Approach to Critical Value Approach to One-Tailed Hypothesis TestingOne-Tailed Hypothesis Testing
The test statistic The test statistic zz has a standard normal probability has a standard normal probability distribution.distribution. We can use the standard normal probabilityWe can use the standard normal probability distribution table to find the distribution table to find the zz-value with an area-value with an area of of aa in the lower (or upper) tail of the distribution. in the lower (or upper) tail of the distribution. The value of the test statistic that established theThe value of the test statistic that established the boundary of the rejection region is called theboundary of the rejection region is called the critical valuecritical value for the test. for the test. The rejection rule is:The rejection rule is:
• Lower tail: Reject Lower tail: Reject HH00 if if zz << - -zz
• Upper tail: Reject Upper tail: Reject HH00 if if zz >> zz
17 17 Slide Slide
00 -za = -1.28 -za = -1.28
Reject H0Reject H0
Do Not Reject H0Do Not Reject H0
zz
Samplingdistribution of
Samplingdistribution of z x
n
0
/z x
n
0
/
Lower-Tailed Test About a Population Mean:Lower-Tailed Test About a Population Mean: σσ KnownKnown
Critical Value ApproachCritical Value Approach Critical Value ApproachCritical Value Approach
18 18 Slide Slide
00 za = 1.645 za = 1.645
Reject H0Reject H0
Do Not Reject H0Do Not Reject H0
zz
Samplingdistribution of
Samplingdistribution of z x
n
0
/z x
n
0
/
Upper-Tailed Test About a Population Mean:Upper-Tailed Test About a Population Mean:σσ Known Known
Critical Value ApproachCritical Value Approach Critical Value ApproachCritical Value Approach
19 19 Slide Slide
Steps of Hypothesis TestingSteps of Hypothesis Testing
Step 1.Step 1. Develop the null and alternative hypotheses. Develop the null and alternative hypotheses.
Step 2.Step 2. Specify the level of significance Specify the level of significance αα..
Step 3.Step 3. Collect the sample data and compute the test Collect the sample data and compute the test statistic.statistic.
pp-Value Approach-Value Approach
Step 4.Step 4. Use the value of the test statistic to compute the Use the value of the test statistic to compute the pp-value.-value.
Step 5.Step 5. Reject Reject HH00 if if pp-value -value << ..
20 20 Slide Slide
Critical Value ApproachCritical Value Approach
Step 4.Step 4. Use the level of significance to determine the Use the level of significance to determine the critical value and the rejection rule.critical value and the rejection rule.
Step 5.Step 5. Use the value of the test statistic and the rejection Use the value of the test statistic and the rejection rule to determine whether to reject rule to determine whether to reject HH00..
Steps of Hypothesis TestingSteps of Hypothesis Testing
21 21 Slide Slide
Example: Metro EMSExample: Metro EMS
The EMS director wants toThe EMS director wants toperform a hypothesis test, with aperform a hypothesis test, with a.05 level of significance, to determine.05 level of significance, to determinewhether the service goal of 12 minutes or less is beingwhether the service goal of 12 minutes or less is beingachieved.achieved.
The response times for a randomThe response times for a randomsample of 40 medical emergenciessample of 40 medical emergencieswere tabulated. The sample meanwere tabulated. The sample meanis 13.25 minutes. The populationis 13.25 minutes. The populationstandard deviation is believed tostandard deviation is believed tobe 3.2 minutes.be 3.2 minutes.
One-Tailed Tests About a Population Mean:One-Tailed Tests About a Population Mean:σσKnownKnown
22 22 Slide Slide
1. Develop the hypotheses.1. Develop the hypotheses.
2. Specify the level of significance.2. Specify the level of significance. = .05= .05
HH00::
HHaa::
pp -Value and Critical Value Approaches -Value and Critical Value Approaches
One-Tailed Tests About a Population Mean:One-Tailed Tests About a Population Mean:σσ Known Known
3. Compute the value of the test statistic.3. Compute the value of the test statistic.
13.25 12 2.47
/ 3.2/ 40x
zn
13.25 12 2.47
/ 3.2/ 40x
zn
23 23 Slide Slide
5. Determine whether to reject 5. Determine whether to reject HH00..
We are at least 95% confident that Metro EMS is We are at least 95% confident that Metro EMS is notnot meeting the response goal of 12 minutes. meeting the response goal of 12 minutes.
pp –Value Approach –Value Approach
One-Tailed Tests About a Population Mean:One-Tailed Tests About a Population Mean:σσ Known Known
4. Compute the 4. Compute the pp –value. –value.
For For zz = 2.47, cumulative probability = .9932. = 2.47, cumulative probability = .9932.
pp–value = 1 - .9932 = .0068–value = 1 - .9932 = .0068
Because Because pp–value = .0068–value = .0068 << = .05, we reject = .05, we reject HH00..
24 24 Slide Slide
p p –Value Approach–Value Approach p p –Value Approach–Value Approach
p-valuep-value
00 za =1.645 za =1.645
= .05 = .05
zz
z =2.47 z =2.47
One-Tailed Tests About a Population Mean:One-Tailed Tests About a Population Mean:σσ Known Known
Samplingdistribution of
Samplingdistribution of z x
n
0
/z x
n
0
/
25 25 Slide Slide
5. Determine whether to reject 5. Determine whether to reject HH00..
We are at least 95% confident that Metro EMS is We are at least 95% confident that Metro EMS is notnot meeting the response goal of 12 minutes. meeting the response goal of 12 minutes.
Because 2.47 Because 2.47 >> 1.645, we reject 1.645, we reject HH00..
Critical Value ApproachCritical Value Approach
One-Tailed Tests About a Population Mean:One-Tailed Tests About a Population Mean:σσKnownKnown
For For = .05, = .05, zz.05.05 = 1.645 = 1.645
4. Determine the critical value and rejection rule.4. Determine the critical value and rejection rule.
Reject Reject HH00 if if zz >> 1.645 1.645
26 26 Slide Slide
pp-Value Approach to-Value Approach toTwo-Tailed Hypothesis TestingTwo-Tailed Hypothesis Testing
The rejection rule:The rejection rule: Reject Reject HH00 if the if the pp-value -value << ..
Compute the Compute the pp-value-value using the following three steps: using the following three steps:
3. Double the tail area obtained in step 2 to obtain3. Double the tail area obtained in step 2 to obtain the the pp –value. –value.
2. If 2. If zz is in the upper tail ( is in the upper tail (zz > 0), find the area under > 0), find the area under the standard normal curve to the right of the standard normal curve to the right of zz.. If If zz is in the lower tail ( is in the lower tail (zz < 0), find the area under < 0), find the area under the standard normal curve to the left of the standard normal curve to the left of zz..
1. Compute the value of the test statistic 1. Compute the value of the test statistic zz..
27 27 Slide Slide
Critical Value Approach to Critical Value Approach to Two-Tailed Hypothesis TestingTwo-Tailed Hypothesis Testing
The critical values will occur in both the lower andThe critical values will occur in both the lower and upper tails of the standard normal curve.upper tails of the standard normal curve.
The rejection rule is:The rejection rule is: Reject Reject HH00 if if zz << - -zz/2/2 or or zz >> zz/2/2..
Use the standard normal probability distributionUse the standard normal probability distribution table to find table to find zz/2/2 (the (the zz-value with an area of -value with an area of aa/2 in/2 in the upper tail of the distribution).the upper tail of the distribution).
28 28 Slide Slide
Example: Glow ToothpasteExample: Glow Toothpaste
Two-Tailed Test About a Population Mean: Two-Tailed Test About a Population Mean: Known Known
oz.
GlowGlow
Quality assurance procedures call forQuality assurance procedures call forthe continuation of the filling process if thethe continuation of the filling process if thesample results are consistent with the assumption thatsample results are consistent with the assumption thatthe mean filling weight for the population of toothpastethe mean filling weight for the population of toothpastetubes is 6 oz.; otherwise the process will be adjusted.tubes is 6 oz.; otherwise the process will be adjusted.
The production line for Glow toothpasteThe production line for Glow toothpasteis designed to fill tubes with a mean weightis designed to fill tubes with a mean weightof 6 oz. Periodically, a sample of 30 tubesof 6 oz. Periodically, a sample of 30 tubeswill be selected in order to check thewill be selected in order to check thefilling process.filling process.
29 29 Slide Slide
Example: Glow ToothpasteExample: Glow Toothpaste
Two-Tailed Test About a Population Mean: Two-Tailed Test About a Population Mean: σσ Known Known
oz.
GlowGlow Perform a hypothesis test, at the .03Perform a hypothesis test, at the .03level of significance, to help determinelevel of significance, to help determinewhether the filling process should continuewhether the filling process should continueoperating or be stopped and corrected.operating or be stopped and corrected.
Assume that a sample of 30 toothpasteAssume that a sample of 30 toothpastetubes provides a sample mean of 6.1 oz.tubes provides a sample mean of 6.1 oz.The population standard deviation is The population standard deviation is believed to be 0.2 oz.believed to be 0.2 oz.
30 30 Slide Slide
1. Determine the hypotheses.1. Determine the hypotheses.
2. Specify the level of significance.2. Specify the level of significance.
3. Compute the value of the test statistic.3. Compute the value of the test statistic.
= .03= .03
pp –Value and Critical Value Approaches –Value and Critical Value Approaches
GloGloww
HH00::
HHaa:: 6 6
Two-Tailed Tests About a Population Mean:Two-Tailed Tests About a Population Mean:σσ Known Known
0 6.1 6
2.74/ .2/ 30
xz
n
0 6.1 6
2.74/ .2/ 30
xz
n
31 31 Slide Slide
GloGloww
Two-Tailed Tests About a Population Mean:Two-Tailed Tests About a Population Mean:σσ Known Known
5. Determine whether to reject 5. Determine whether to reject HH00..
pp –Value Approach –Value Approach
4. Compute the 4. Compute the pp –value. –value.
For For zz = 2.74, cumulative probability = .9969 = 2.74, cumulative probability = .9969
pp–value = 2(1 - .9969) = .0062–value = 2(1 - .9969) = .0062
Because Because pp–value = .0062 –value = .0062 << = = .03, we reject .03, we reject HH00..
We are at least 97% confident that the mean filling We are at least 97% confident that the mean filling weight of the toothpaste tubes is not 6 oz.weight of the toothpaste tubes is not 6 oz.
32 32 Slide Slide
GloGloww
Two-Tailed Tests About a Population Mean:Two-Tailed Tests About a Population Mean:σσ Known Known
/2 = .015/2 = .015
00za/2 = 2.17za/2 = 2.17
zz
/2 = .015/2 = .015
pp-Value Approach-Value Approach
-za/2 = -2.17-za/2 = -2.17z = 2.74z = 2.74z = -2.74z = -2.74
1/2p -value= .0031
1/2p -value= .0031
1/2p -value= .0031
1/2p -value= .0031
33 33 Slide Slide
Critical Value ApproachCritical Value Approach
GloGloww
Two-Tailed Tests About a Population Mean:Two-Tailed Tests About a Population Mean:σσ Known Known
5. Determine whether to reject 5. Determine whether to reject HH00..
We are at least 97% confident that the mean filling We are at least 97% confident that the mean filling weight of the toothpaste tubes is not 6 oz.weight of the toothpaste tubes is not 6 oz.
Because 2.47 Because 2.47 >> 2.17, we reject 2.17, we reject HH00..
For For aa/2 = .03/2 = .015, /2 = .03/2 = .015, zz.015.015 = 2.17 = 2.17
4. Determine the critical value and rejection rule.4. Determine the critical value and rejection rule.
Reject Reject HH00 if if zz << -2.17 or -2.17 or zz >> 2.17 2.17
34 34 Slide Slide
a/2 = .015a/2 = .015
00 2.17 2.17
Reject H0Reject H0Do Not Reject H0Do Not Reject H0
zz
Reject H0Reject H0
-2.17 -2.17
GloGloww
Critical Value ApproachCritical Value Approach
Samplingdistribution of
Samplingdistribution of z x
n
0
/z x
n
0
/
Two-Tailed Tests About a Population Mean:Two-Tailed Tests About a Population Mean:σσ Known Known
a/2 = .015a/2 = .015
35 35 Slide Slide
Confidence Interval Approach toConfidence Interval Approach toTwo-Tailed Tests About a Population MeanTwo-Tailed Tests About a Population Mean
Select a simple random sample from the populationSelect a simple random sample from the population and use the value of the sample mean to developand use the value of the sample mean to develop the confidence interval for the population mean the confidence interval for the population mean μμ.. (Confidence intervals are covered in Chapter 8.)(Confidence intervals are covered in Chapter 8.)
xx
If the confidence interval contains the hypothesizedIf the confidence interval contains the hypothesized valuevalue 00, do not reject , do not reject HH00. Otherwise, reject . Otherwise, reject HH00..
36 36 Slide Slide
The 97% confidence interval for The 97% confidence interval for μμisis
/ 2 6.1 2.17(.2 30) 6.1 .07924x zn
/ 2 6.1 2.17(.2 30) 6.1 .07924x zn
Confidence Interval Approach toConfidence Interval Approach toTwo-Tailed Tests About a Population MeanTwo-Tailed Tests About a Population Mean
GloGloww
Because the hypothesized value for theBecause the hypothesized value for thepopulation mean,population mean, 00 = 6, is not in this interval,= 6, is not in this interval,the hypothesis-testing conclusion is that thethe hypothesis-testing conclusion is that thenull hypothesis, null hypothesis, HH00:: = 6, can be rejected.= 6, can be rejected.
or 6.02076 to 6.17924or 6.02076 to 6.17924
37 37 Slide Slide
Test StatisticTest Statistic
9.4 Tests About a Population Mean:9.4 Tests About a Population Mean: Unknown Unknown
txs n
0/
txs n
0/
This test statistic has a This test statistic has a tt distribution distribution with with nn - 1 degrees of freedom. - 1 degrees of freedom.
38 38 Slide Slide
Rejection Rule: Rejection Rule: pp -Value Approach -Value Approach
HH00:: Reject Reject HH0 0 if if tt >> tt
Reject Reject HH0 0 if if tt << - -tt
Reject Reject HH0 0 if if tt << - - tt or or tt >> tt
HH00::
HH00::
Tests About a Population Mean:Tests About a Population Mean: Unknown Unknown
Rejection Rule: Critical Value ApproachRejection Rule: Critical Value Approach
Reject Reject HH0 0 if if p p –value–value <<
39 39 Slide Slide
p p -Values and the -Values and the tt Distribution Distribution
The format of the The format of the tt distribution table provided in most distribution table provided in most statistics textbooks does not have sufficient detailstatistics textbooks does not have sufficient detail to determine the to determine the exactexact p p-value for a hypothesis test.-value for a hypothesis test.
However, we can still use the However, we can still use the tt distribution table to distribution table to identify a identify a rangerange for the for the pp-value.-value.
An advantage of computer software packages is thatAn advantage of computer software packages is that the computer output will provide the the computer output will provide the pp-value for the-value for the tt distribution. distribution.
40 40 Slide Slide
A State Highway Patrol periodically samplesA State Highway Patrol periodically samplesvehicle speeds at various locationsvehicle speeds at various locationson a particular roadway. on a particular roadway. The sample of vehicle speedsThe sample of vehicle speedsis used to test the hypothesisis used to test the hypothesis
Example: Highway PatrolExample: Highway Patrol
One-Tailed Test About a Population Mean: One-Tailed Test About a Population Mean: Unknown Unknown
The locations where The locations where HH00 is rejected are deemed is rejected are deemed
the best locations for radar traps.the best locations for radar traps.
HH00:: << 65 65
41 41 Slide Slide
Example: Highway PatrolExample: Highway Patrol
One-Tailed Test About a Population Mean: One-Tailed Test About a Population Mean: σσ Unknown Unknown
At Location F, a sample of 64 vehicles shows aAt Location F, a sample of 64 vehicles shows amean speed of 66.2 mph with amean speed of 66.2 mph with astandard deviation ofstandard deviation of4.2 mph. Use 4.2 mph. Use αα = .05 to = .05 totest the hypothesis.test the hypothesis.
42 42 Slide Slide
One-Tailed Test About a Population Mean:One-Tailed Test About a Population Mean:σσ Unknown Unknown
1. Determine the hypotheses.1. Determine the hypotheses.
2. Specify the level of significance.2. Specify the level of significance.
3. Compute the value of the test statistic.3. Compute the value of the test statistic.
= .05= .05
pp –Value and Critical Value Approaches –Value and Critical Value Approaches
HH00:: << 65 65
HHaa:: > 65 > 65
0 66.2 65
2.286/ 4.2/ 64x
ts n
0 66.2 65
2.286/ 4.2/ 64x
ts n
43 43 Slide Slide
One-Tailed Test About a Population Mean:One-Tailed Test About a Population Mean:σσ Unknown Unknown
pp –Value Approach –Value Approach
5. Determine whether to reject 5. Determine whether to reject HH00..
4. Compute the 4. Compute the pp –value. –value.
For For tt = 2.286, the = 2.286, the pp–value must be less than .025–value must be less than .025(for (for tt = 1.998) and greater than .01 (for = 1.998) and greater than .01 (for tt = 2.387). = 2.387).
.01 < .01 < pp–value < .025–value < .025
Because Because pp–value–value << = .05, we reject = .05, we reject HH00..
We are at least 95% confident that the mean speedWe are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph.of vehicles at Location F is greater than 65 mph.
44 44 Slide Slide
Critical Value ApproachCritical Value Approach
5. Determine whether to reject 5. Determine whether to reject HH00..
We are at least 95% confident that the mean speed of We are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph. Location vehicles at Location F is greater than 65 mph. Location F is a good candidate for a radar trap.F is a good candidate for a radar trap.
Because 2.286 Because 2.286 >> 1.669, we reject 1.669, we reject HH00..
One-Tailed Test About a Population Mean:One-Tailed Test About a Population Mean:σσ Unknown Unknown
For For aa = .05 and d.f. = 64 – 1 = 63, = .05 and d.f. = 64 – 1 = 63, tt.05.05 = 1.669 = 1.669
4. Determine the critical value and rejection rule.4. Determine the critical value and rejection rule.
Reject Reject HH00 if if tt >> 1.669 1.669
45 45 Slide Slide
00 ta =1.669 ta =1.669
Reject H0Reject H0
Do Not Reject H0Do Not Reject H0
tt
One-Tailed Test About a Population Mean:One-Tailed Test About a Population Mean:σσ UnknownUnknown
46 46 Slide Slide
The equality part of the hypotheses always appearsThe equality part of the hypotheses always appears in the null hypothesis.in the null hypothesis. In general, a hypothesis test about the value of aIn general, a hypothesis test about the value of a population proportion population proportion pp must take one of themust take one of the
following three forms (where following three forms (where pp00 is the hypothesized is the hypothesized
value of the population proportion).value of the population proportion).
9.5 A Summary of Forms for Null and Alternative 9.5 A Summary of Forms for Null and Alternative Hypotheses About a Population ProportionHypotheses About a Population Proportion
One-tailedOne-tailed(lower tail)(lower tail)
One-tailedOne-tailed(upper tail)(upper tail)
Two-tailedTwo-tailed
0 0: H p p0 0: H p p
0: aH p p 0: aH p p0: aH p p 0: aH p p0 0: H p p0 0: H p p 0 0: H p p0 0: H p p
0: aH p p 0: aH p p
47 47 Slide Slide
Test StatisticTest Statistic
zp p
p
0
z
p p
p
0
pp p
n 0 01( ) pp p
n 0 01( )
Tests About a Population ProportionTests About a Population Proportion
where:where:
assuming assuming npnp >> 5 and 5 and nn(1 – (1 – pp) ) >> 5 5
48 48 Slide Slide
Rejection Rule: Rejection Rule: pp –Value Approach –Value Approach
HH00: : pppp Reject Reject HH0 0 if if zz >> zzαα
Reject Reject HH0 0 if if zz << - -zz
Reject Reject HH0 0 if if zz << - -zz or or zz >> zz
HH00: : pppp
HH00: : pppp
Tests About a Population ProportionTests About a Population Proportion
Reject Reject HH0 0 if if p p –value –value << aa
Rejection Rule: Critical Value ApproachRejection Rule: Critical Value Approach
49 49 Slide Slide
Example: National Safety CouncilExample: National Safety Council For a Christmas and New Year’s week, theFor a Christmas and New Year’s week, theNational Safety Council estimated thatNational Safety Council estimated that500 people would be killed and 25,000500 people would be killed and 25,000injured on the nation’s roads. Theinjured on the nation’s roads. TheNSC claimed that 50% of theNSC claimed that 50% of theaccidents would be caused byaccidents would be caused bydrunk driving.drunk driving.
Two-Tailed Test About aTwo-Tailed Test About aPopulation ProportionPopulation Proportion
50 50 Slide Slide
A sample of 120 accidents showed thatA sample of 120 accidents showed that67 were caused by drunk driving. Use67 were caused by drunk driving. Usethese data to test the NSC’s claim withthese data to test the NSC’s claim withaa = .05. = .05.
Two-Tailed Test About aTwo-Tailed Test About aPopulation ProportionPopulation Proportion
Example: National Safety CouncilExample: National Safety Council
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Two-Tailed Test About aTwo-Tailed Test About aPopulation ProportionPopulation Proportion
1. Determine the hypotheses.1. Determine the hypotheses.
2. Specify the level of significance.2. Specify the level of significance.
3. Compute the value of the test statistic.3. Compute the value of the test statistic.
αα = .05 = .05
pp –Value and Critical Value Approaches –Value and Critical Value Approaches
0: .5H p: .5aH p
0 (67/ 120) .5 1.28
.045644p
p pz
0 (67/ 120) .5 1.28
.045644p
p pz
0 0(1 ) .5(1 .5).045644
120p
p p
n
0 0(1 ) .5(1 .5)
.045644120p
p p
n
a commona commonerror is usingerror is using
in this in this formula formula pp
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pp-Value Approach-Value Approach
4. Compute the 4. Compute the pp -value. -value.
5. Determine whether to reject 5. Determine whether to reject HH00..
Because Because pp–value = .2006 > –value = .2006 > aa = .05, we cannot reject = .05, we cannot reject HH00..
Two-Tailed Test About aTwo-Tailed Test About aPopulation ProportionPopulation Proportion
For For zz = 1.28, cumulative probability = .8997 = 1.28, cumulative probability = .8997
pp–value = 2(1 - .8997) = .2006–value = 2(1 - .8997) = .2006
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Two-Tailed Test About aTwo-Tailed Test About aPopulation ProportionPopulation Proportion
Critical Value ApproachCritical Value Approach
5. Determine whether to reject 5. Determine whether to reject HH00..
For For aa/2 = .05/2 = .025, /2 = .05/2 = .025, zz.025.025 = 1.96 = 1.96
4. Determine the criticals value and rejection rule.4. Determine the criticals value and rejection rule.
Reject Reject HH00 if if zz << -1.96 or -1.96 or zz >> 1.96 1.96
Because 1.278 > -1.96 and < 1.96, we cannot reject Because 1.278 > -1.96 and < 1.96, we cannot reject HH00..
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HomeworkHomework
Exercises: 5, 10, 11, 15, 24, 27, 36, 38Exercises: 5, 10, 11, 15, 24, 27, 36, 38
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End of Chapter 9End of Chapter 9