Unformatted text preview: BIT 2405 Quantitative Methods I
BIT 2405 Quantitative Methods I
Chapter 9a
Hypothesis Tests This Week
This Week
Lectures Help Session 11am
Help
Only.
Only.
GO HOKIES! Hawkes Today
HW911
HW911
and
REQUIRE
REQUIRE
D Quiz 3
will be
will
posted this
weekend.
weekend.
I’ll Answer Questions on TEST 2 in the Help Sessions
I’ll
Today and Thursday Morning.
Today Next Week and Beyond.
Next Week and Beyond.
Lectures Hawkes
I suggest you
suggest
work on Hawkes
EARLY!
EARLY!
9.1 is not required
9.1
but you may want
to do the instruct
as an Intro to
Hypothesis
Testing
Testing If You Struggled on Test 2, Get It Figured If You Struggled on Test 2, Out NOW, Because We Build on These Concepts from Here On!
• Come to the Help Sessions
– ASK QUESTIONS! • Hypothesis Testing BUILDS on the Material in Test 2.
–
–
– Clearly knowing values versus areas/probabilities.
Looking up values or areas.
Keeping straight what you can look up versus what answers the question. • Figure out a better TEST TAKING Strategy if you had trouble finishing. Questions?
Questions? Chapter 9 & 11.1
Chapter 9 & 11.1 Hypothesis Tests
This Week
KEY
Chapter TEXT
Quantitative Methods I,
Anderson, Sweeney & Williams Today Section Topic 15
Ch09a.p
pt 9.1 Hawkes Learning Systems:
Statistics
Certifications
Section Topic 9.2 9.3
9.4 T X Type I and Type II
Errors
Population Mean: σ
Known
Population Mean: σ Y ou need to UNDERSTAND
these so you can make
Hypothesis Testing
sense of the problems.
sense
Proportions: P Value X Developing Null and
Alternative
Hypotheses
9.2 16
Ch09b.p
pt T Certifications due
Certifications
M onday, November
9th
@ 11:59pm
11:59pm 9.3 Hypothesis Testing
Proportions: z Value 9.4 Hypothesis Testing
Means: P Value
Hypothesis Testing 9.5 Chapter 9
Chapter 9 Hypothesis Testing Developing Null and Alternative Hypotheses Type I and Type II Errors Population Mean: σ Known Population Mean: σ Unknown Population Proportion How Would You Approach This Question Now?
How Would You Approach This Question Now? During the Test, It was really HOT in the During the Test, It was really HOT in the room.
How hot was it?
I’d bet it was AT LEAST 75 degrees! I’d bet it was AT LEAST 75 degrees!
I’d bet it was AT LEAST 75 degrees!
A guess about µ
µ ≥75 H0: µ ≥75 I’d bet it was AT LEAST 75 degrees!
I’d bet it was AT LEAST 75 degrees!
A guess about µ
µ ≥75 H0: µ ≥75 Ha: µ < 75 I’d bet it was AT LEAST 75 degrees!
I’d bet it was AT LEAST 75 degrees!
H0: µ ≥75 We will conduct a TEST of the Null Hypothesis.
Collect a sample and look at the sample mean x̅obs Ha: µ < 75 We call the
We
hypothesized
mean µ0 so
mean
µ0 = 75 With the TEST STATISTIC (from the sample),
we can either REJECT or NOT REJECT the
we
Null Hypothesis.
z = xobs − µ
obs σ/ n 0 1
σx̅ z=0
z=0
μ0=75 x̅ I’d bet it was AT LEAST 75 degrees!
I’d bet it was AT LEAST 75 degrees!
H0: µ ≥75 We will conduct a TEST of the Null Hypothesis.
Collect a sample and look at the sample mean x̅obs Ha: µ < 75
With the TEST STATISTIC (from the sample),
we can either REJECT or NOT REJECT the
we
Null Hypothesis.
z = xobs − µ
obs Level of Significance: α Z
critical z=0
μ0=75 σ/ n 0 I’d bet it was AT LEAST 75 degrees!
I’d bet it was AT LEAST 75 degrees!
H0: µ is ≥75 We will conduct a TEST of the Null Hypothesis.
Collect a sample and look at the sample mean x̅obs Ha: µ is < 75
With the TEST STATISTIC (from the sample),
we can either REJECT or NOT REJECT the
we
Null Hypothesis.
z = xobs − µ
obs If zobs i s outside the critical
value, we reject the
value,
null hypothesis
null
reject Z zobs critical z=0
μ0=75 σ/ n 0 I’d bet it was AT LEAST 75 degrees!
I’d bet it was AT LEAST 75 degrees!
H0: µ is ≥75 We will conduct a TEST of the Null Hypothesis.
Collect a sample and look at the sample mean x Ha: µ is < 75
With the TEST STATISTIC (from the sample),
we can either REJECT or NOT REJECT the
we
Null Hypothesis.
z = x−µ
obs If zobs i s NOT outside the critical
value, we DO NOT reject the
value,
null hypothesis
null
do not reject Z
critical zobs z=0
μ0=75 0 σ/ n Developing Hypotheses
Developing Hypotheses ? Start with a QUESTION.
“ How do we know she’ s a
How
witch?”
witch?” Population L ook for Characteristics of
Look
Elements we can measure and a
plausible explanation that can
be tested.
be
“ I f she weighs the same as
If
a duck, she’ s a witch.”
duck,
H0: µ = µ0 weightofduck
weightofduck
Ha: µ ≠µ0 weightofduck Testing Hypotheses
Testing Hypotheses
H0 Null Hypothesis ? Ha1 , Ha2 ... Alternative Hypotheses
Populationst
Te
•
•
••
• • •
• le
p
m
Sa •
• Developing Null and Alternative Hypotheses
Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected. The null hypothesis, denoted by H0 , is a tentative assumption about a population parameter. The alternative hypothesis, denoted by Ha, is the opposite of what is stated in the null hypothesis. The alternative hypothesis is what the test is attempting to establish. The Text Discusses 3 Types of Hypotheses
The Text Discusses 3 Types of Hypotheses
s
s
s Research Hypotheses
Claim Hypotheses
DecisionMaking Hypotheses Hawkes Tends to Establish the Null Hypothesis According to the Text’s Research Approach to Hypotheses Developing Null and Alternative Hypotheses
Developing Null and Alternative Hypotheses
s Testing Research Hypotheses • The research hypothesis should be expressed as the alternative hypothesis. • The conclusion that the research hypothesis is true comes from sample data that contradict the null hypothesis. Developing Null and Alternative Hypotheses
Developing Null and Alternative Hypotheses
s Testing the Validity of a Claim • Manufacturers’ claims are usually given the benefit of the doubt and stated as the null hypothesis.
• The conclusion that the claim is false comes from sample data that contradict the null hypothesis. Developing Null and Alternative Hypotheses
Developing Null and Alternative Hypotheses
s Testing in DecisionMaking Situations • A decision maker might have to choose between two courses of action, one associated with the null hypothesis and another associated with the alternative hypothesis. • Example: Accepting a shipment of goods from a supplier or returning the shipment of goods to the supplier Summary of Forms for Null and Alternative Hypotheses about a Population Mean
s The equality part of the hypotheses always appears in the null hypothesis. In general, an hypothesis test about the value of a population mean µ must take one of the following
must take one of the following three forms (where µ 0 is the hypothesized value of the population mean). H 0 : µ ≥ µ0
H a : µ < µ0 H 0 : µ ≤ µ0
H a : µ > µ0 H 0 : µ = µ0
H a : µ ≠ µ0 Onetailed
Onetailed
(lowertail) Onetailed
(uppertail) Twotailed Hypothesis Testing
Hypothesis Testing
Three Possible Forms The objective of the test is to REJECT the
The
claim of the Null Hypothesis.
claim H0: μ ≥µ0 H0: μ ≤µ0 H0: μ = µ0 Ha: μ < µ0 Ha: μ > µ0 Ha: μ ≠µ0 Hypothesis Testing
Hypothesis Testing
Three Possible Forms The objective of the test is to REJECT the
The
claim of the Null Hypothesis.
claim H0: μ ≥µ0 H0: μ ≤µ0 H0: μ = µ0 Ha: μ < µ0 Ha: μ > µ0 Ha: μ ≠µ0 Hypothesis Testing
Three Possible Forms
reject H0: μ ≥µ0 Ha: μ < µ0 The objective of the test is to REJECT the
The
claim of the Null Hypothesis.
claim
reject
reject H0: μ ≤µ0 H0: μ = µ0 Ha: μ > µ0 Ha: μ ≠µ0 So in this case, our objective is see if the test sample mean (x̅1)
i ndicates the sample is from a SIGNIFICANTLY DIFFERENT
population with a mean μ that is LESS THAN the claimed value
μ0 rather than from the claimed population with μ ≥μ0 . Null and Alternative Hypotheses
Null and Alternative Hypotheses
s Example: Metro EMS A major west coast city provides one of the most
comprehensive emergency medical services in the
world. Operating in a multiple hospital system
with approximately 20 mobile medical units, the
service goal is to respond to medical emergencies
with a mean time of 12 minutes or less. The director of medical services wants to
formulate a hypothesis test that could use a sample
of emergency response times to determine whether
or not the service goal of 12 minutes or less is being
achieved. Null and Alternative Hypotheses
Null and Alternative Hypotheses
H0: µ < 1 2 Ha: µ > 1 2 The emergency service is meeting
the response goal; no followup
action is necessary.
The emergency service is not
meeting the response goal;
appropriate followup action is
necessary. where: µ = mean response time for the population of medical emergency requests Type I Error
Type I Error Because hypothesis tests are based on sample data, we must allow for the possibility of errors.
s A Type I error is rejecting H0 when it is true. s The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance.
s Applications of hypothesis testing that only control the Type I error are often called significance tests. Type II Error
Type II Error
s A Type II error is accepting H0 when it is false.
s It is difficult to control for the probability of making a Type II error.
s Statisticians avoid the risk of making a Type II error by using “do not reject H0” and not “accept H0”. Type I and Type II Errors
Type I and Type II Errors
Population Condition Conclusion H0 True
(µ < 12) H0 False
(µ > 12) Accept H0
(Conclude µ < 12) Correct
Decision Type II Error Type I Error Correct
Decision Reject H0
(Conclude µ > 12) There Are Two Approaches to Testing the Null There Are Two Approaches to Testing the Null Hypothesis.
s pValue Approach
• Determine the test statistic
• Determine the pvalue which is an area to the left or right of the test statistic and in some cases 2 x area.
• Compare the pvalue to α.
• Determine to reject or not reject the null hypothesis. s Critical Value Approach
• Use α to find the critical value (a value with α to the left or right or α/2 in some cases)
• Compare the test statistic to the critical value.
• Determine to reject or not reject the null hypothesis. pValue Approach to
OneTailed Hypothesis Testing The pvalue is the probability, computed using the test statistic, that measures the support (or lack of support) provided by the sample for the null hypothesis. If the pvalue is less than or equal to the level of significance α, the value of the test statistic is in the rejection region. Reject H0 if the pvalue < α . pvalue
value
Think PROBABILITY
Think
pvalue is a probability (area under the curve) LowerTailed Test About a Population Mean:
σ Known
s pValue Approach pValue < α ,
so reject H0. α = .10 Sampling
distribution
x−µ of zobs = σ / n
0 pvalue
= .0 7 2 z z = zα = 1.28
obs
1.46 0 UpperTailed Test About a Population Mean:
σ Known
s pValue Approach pValue < α ,
so reject H0. Sampling
distribution
x−µ
z of obs = σ / n α = .04 0 pValue
= .0 1 1
z
0 zα = 1.75 z =
obs 2.29 Critical Value Approach to Critical Value Approach to OneTailed Hypothesis Testing The test statistic zobs has a standard normal probability distribution. We can use the standard normal probability distribution table to find the zvalue with an area of α in the lower (or upper) tail of the distribution. The value of the test statistic that established the boundary of the rejection region is called the critical value for the test.
s The rejection rule is: • Lower tail: Reject H0 if zobs < zα
• Upper tail: Reject H0 if zobs > zα LowerTailed Test About a Population Mean:
σ Known
s Critical Value Approach Sampling
distribution
zobs = x − µ of σ / n
0 Reject H0 α = .1 0 − α = −
z
1.28 Do Not Reject H0
0 z UpperTailed Test About a Population Mean:
σ Known
s Critical Value Approach Sampling
distribution
x−µ
z of obs = σ / n
0 Reject H0 Do Not Reject H0
0 α = .0 5 zα = 1.645 z Steps of Hypothesis Testing
Steps of Hypothesis Testing
Step 1. Develop the null and alternative hypotheses.
Step 2. Specify the level of significance α.
Step 3. Collect the sample data and compute the test statistic.
pValue Approach
Step 4. Use the value of the test statistic to compute the pvalue.
Step 5. Reject H0 if pvalue < α. Steps of Hypothesis Testing
Critical Value Approach
Critical Value Approach
Step 4. Use the level of significance to determine the to determine the critical value and the rejection rule.
Step 5. Use the value of the test statistic and the rejection rule to determine whether to reject H0. OneTailed Tests About a Population Mean:
σ Known
s Example: Metro EMS The response times for a random sample of 40
medical emergencies were tabulated. The sample
mean is 13.25 minutes. The population standard
deviation is believed to be 3.2 minutes. The EMS director wants to perform a hypothesis
test, with a .05 level of significance, to determine
whether the service goal of 12 minutes or less is being achieved. OneTailed Tests About a Population Mean:
σ Known p Value and Critical Value Approaches 1. Develop the hypotheses.
1. Develop the hypotheses. H0: µ < 1 2
Ha: µ > 1 2 2. Specify the level of significance. α = .05 3. Compute the value of the test statistic. z obs x − µ 0 13.25 − 12
=
=
= 2.47
σ / n 3.2 / 40 OneTailed Tests About a Population Mean:
σ Known p –Value Approach 4. Compute the p –value.
For zobs = 2.47, cumulative probability = .9932.
p–value = 1 − .9932 = .0068
5. Determine whether to reject H0.
5. Determine whether to reject Because p–value = .0068 < α = .05, we reject H0.
There is sufficient statistical evidence
to infer that Metro EMS is not meeting
the response goal of 12 minutes. OneTailed Tests About a Population Mean:
σ Known
s p –Value Approach Sampling
distribution
x−µ
z of obs = σ / n α = .05 0 pvalue
= .0 0 6 8
z
0 zα =
1.645 z =
obs 2.47 OneTailed Tests About a Population Mean:
σ Known Critical Value Approach 4. Determine the critical value and rejection rule.
For α = .05, z.05 = 1.645
Reject H0 if zobs > 1.645
5. Determine whether to reject H0.
5. Determine whether to reject Because 2.47 > 1.645, we reject H0.
There is sufficient statistical evidence
to infer that Metro EMS is not meeting
the response goal of 12 minutes. OneTailed Tests About a Population Mean:
σ Known
s p –Value Approach Sampling
distribution
x−µ
z of obs = σ / n
0 z
0 zα =
1.645 z =
obs 2.47 pValue Approach to
TwoTailed Hypothesis Testing Compute the pvalue using the following three steps: 1. Compute the value of the test statistic zobs. 2. If zobs is in the upper tail (zobs > 0), find the area under the standard normal curve to the right of zobs. If zobs is in the lower tail (zobs < 0), find the area under the standard normal curve to the left of zobs.
3. Double the tail area obtained in step 2 to obtain the p –value. The rejection rule: Reject H0 if the pvalue < α . Critical Value Approach to Critical Value Approach to TwoTailed Hypothesis Testing
he critical values will occur in both the lower and
pper tails of the standard normal curve. Use the standard normal probability distribution table to find zα/2 (the zvalue with an area of α /2 in the upper tail of the distribution).
s The rejection rule is: Reject H0 if zobs < zα/2 or zobs > zα/2. TwoTailed Tests About a Population Mean:
σ Known
s Example: Glow Toothpaste The production line for Glow toothpaste is
designed to fill tubes with a mean weight of 6 oz.
Periodically, a sample of 30 tubes will be selected in
order to check the filling process. Quality assurance procedures call for the
continuation of the filling process if the sample
results are consistent with the assumption that the
mean filling weight for the population of toothpaste
tubes is 6 oz.; otherwise the process will be adjusted. TwoTailed Tests About a Population Mean:
σ Known
s Example: Glow Toothpaste Assume that a sample of 30 toothpaste tubes
provides a sample mean of 6.1 oz. The population
standard deviation is believed to be 0.2 oz. Perform a hypothesis test, at the .03 level of significance, to help determine whether the filling
process should continue operating or be stopped and
corrected. TwoTailed Tests About a Population Mean:
σ Known p –Value and Critical Value Approaches 1. Determine the hypotheses.
1. Determine the hypotheses. H0: µ = 6
Ha: µ ≠ 6 2. Specify the level of significance. α = .03 3. Compute the value of the test statistic. z obs x − µ0
6.1 − 6
=
=
= 2.74
σ / n .2 / 30 TwoTailed Tests About a Population Mean:
TwoTailed Tests About a Population Mean:
σ Known p –Value Approach 4. Compute the p –value.
For zobs = 2.74, cumulative probability = .9969
p–value = 2(1 − .9969) = .0062
5. Determine whether to reject H0.
Because p–value = .0062 < α = .03, we reject H0.
There is sufficient statistical evidence to
infer that the alternative hypothesis is true (i.e. the mean filling weight is not 6 ounces). TwoTailed Tests About a Population Mean:
TwoTailed Tests About a Population Mean:
σ Known pValue Approach 1/2
p value
= .0031 1/2
p value
= .0031 α /2 = .015 α /2 = .015
z = 2.74
zα/2 = 2.17
obs z 0 zα/2 = 2.17 z = 2.74
obs TwoTailed Tests About a Population Mean:
σ Known Critical Value Approach 4. Determine the critical value and rejection rule.
For α /2 = .03/2 = .015, z.015 = 2.17
Reject H0 if zobs < 2.17 or zobs > 2.17
5. Determine whether to reject H0.
Because 2.74 > 2.17, we reject H0.
There is sufficient statistical evidence to
infer that the alternative hypothesis is true (i.e. the mean filling weight is not 6 ounces). TwoTailed Tests About a Population Mean:
σ Known Critical Value Approach Sampling
distribution
x−µ of zobs = σ / n
0 Reject H0 Reject H0 Do Not Reject H0 α/2 = .015 2.17 α/2 = .015
0 2.17 z Confidence Interval Approach to
Confidence Interval Approach to
TwoTailed Tests About a Population Mean Select a simple random sample from the population x and use the value of the sample mean to develop the confidence interval for the population mean µ . (Confidence intervals are covered in Chapter 8.) If the confidence interval contains the hypothesized value µ 0, do not reject H0. Otherwise, reject H0. Confidence Interval Approach to
Confidence Interval Approach to
TwoTailed Tests About a Population Mean The 97% confidence interval for µ is x ± zα / 2 σ
= 6.1 ± 2.17(.2
n 30) = 6.1 ± .07924 or 6.02076 to 6.17924 Because the hypothesized value for the
population mean, µ 0 = 6, is not in this interval,
the hypothesistesting conclusion is that the
null hypothesis, H0: µ = 6, can be rejected. Questions?
Questions? ...
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 Summer '03
 DHBannan
 Null hypothesis, Statistical hypothesis testing, alternative hypotheses, Zobs

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