Hypothesis Testing: Deciding between Reality and Coincidence

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LESSON 3

• Hypothesis Testing Deciding between Reality and
  Coincidence

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Hypothesis Testing

• Developing Null and Alternative Hypotheses
• Type I and Type II Errors
• One-Tailed Tests About a Population Mean
• Two-Tailed Tests About a Population Mean
• Tests About a Population Proportion
• Hypothesis Testing and Decision Making
• Tests About a Population Proportion

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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. It is what we want to prove.
• Hypothesis testing is similar to a criminal
  trial. The hypotheses are
• H0 The defendant is innocent
• Ha The defendant is guilty

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Developing Null and Alternative Hypotheses

• 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.
• Alternative What you want to prove
• Null Status Quo

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Developing Null and Alternative Hypotheses

• 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.

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Type I and Type II Errors

• Since hypothesis tests are based on sample data,
  we must allow for the possibility of errors.
• A Type I error is rejecting H0 when it is true.
• A Type II error is accepting H0 when it is false.
• The person conducting the hypothesis test
  specifies the maximum allowable probability of
  making a
• Type I error, denoted by ? and called the level
  of significance.
• It is difficult to control for the probability
  of making
• a Type II error.
• Type I error Sending an innocent to jail
• Type II error freeing a guilty defendant.

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Summary of Forms for Null and Alternative
Hypotheses about a Population Mean

• The equality part of the hypotheses always
  appears
• in the null hypothesis.

• In general, a hypothesis test about the value
  of a
• population mean ?? must take one of the
  following
• three forms (where ?0 is the hypothesized
  value of
• the population mean).

One-tailed (lower-tail)
One-tailed (upper-tail)
Two-tailed
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Example Metro EMS

• Null and Alternative Hypotheses
• 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.

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Example Metro EMS

• Null and Alternative Hypotheses
• Hypotheses Conclusion and Action
• H0 ?????? The emergency service is
  meeting
• the response goal no follow-up
• action is necessary.
• Ha???????? The emergency service is
  not
• meeting the response goal
• appropriate follow-up action is
• necessary.
• Where ? mean response time for the
  population
• of medical emergency
  requests.

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Example Metro EMS

• Type I and Type II Errors
• Population Condition
• H0 True Ha True
• Conclusion (?????? ) (?????? )
• Accept H0 Correct Type II
• (Conclude ???????
  Conclusion Error
• Reject H0 Type
  I Correct
• (Conclude ??????? ???? ??rror
  Conclusion

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Level Of Significance VS Level Of Confidence

• The level of confidence indicates the probability
  that the alternative hypothesis is correct if the
  null hypothesis is rejected. A 90 level of
  confidence can also be stated as a 10 level of
  significance, and results can be summarized by
  saying that a coefficient has been shown to be
  statistically significant at the 10 level of
  significance or the 90 level of confidence.
• 1 -? is the confidence coefficient
• ? is the level of significance

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The Steps of Hypothesis Testing

• 1. Determine the appropriate hypotheses.
• 2. Select the test statistic for deciding
  whether or not to reject the null hypothesis.
• 3. Specify the level of significance ? for the
  test or the level of confidence 1- ?.
• 4. Use ? or 1- ? to develop the rule for
  rejecting H0.
• 5. Collect the sample data and compute the value
  of the test statistic.

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Three methods to test the Null

• a) Compare the test statistic to the critical
  value(s) in the rejection rule, or
• b) Compute the p-value based on the test
  statistic and compare it to ???to determine
  whether or not to reject H0, or
• c) Build a Confidence Interval If the confidence
  interval contains the hypothesized value ?0, do
  not reject H0. Otherwise, reject H0.

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First Method One-Tailed Tests about a Population
Mean s Known

• Hypotheses
• H0 ?????? ? or H0 ??????
• Ha???????? ?Ha????????
• Test Statistic ?? Known

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Continued

• Rejection Rule

H0 ?????? Reject H0 if z gt
z? H0 ?????? Reject H0 if z lt -z?
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Upper-Tailed Test About a Population Mean

• Critical Value Approach

Reject H0
???????
Do Not Reject H0
z
za 1.645
0
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Lower-Tailed Test About a Population Mean

• Critical Value Approach

Reject H0
a ???1?
Do Not Reject H0
z
-za -1.28
0
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Second Method The Use of p-Values

• The p-value is the probability of obtaining a
  sample result that is at least as unlikely as
  what is observed.
• The p-value can be used to make the decision in a
  hypothesis test by noting that
• if the p-value is less than the level of
  significance ?, the value of the test statistic
  is in the rejection region.
• if the p-value is greater than or equal to ?, the
  value of the test statistic is not in the
  rejection region.
• Reject H0 if the p-value lt ?.

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Lower-Tailed Test About a Population Mean
p-Value lt a , so reject H0.

• p-Value Approach

a .10
p-value ????72
z
0
-za -1.28
z -1.46
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Upper-Tailed Test About a Population Mean
p-Value lt a , so reject H0.

• p-Value Approach

a .04
p-Value ????11
z
za 1.75
z 2.29
0
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Third Method One-Tailed Confidence Interval

• We are sure that the population mean is at
  least as large as (-)
• Or We are sure that the population mean is not
  larger than ()
• If CI does not include ? Reject Null

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Example Metro EMS

• One-Tailed Test about a Population Mean Large n
• Let ? P(Type I Error) .05

Sampling distribution of (assuming H0 is
true and ? 12)
Reject H0
Do Not Reject H0
???????
1.645?
c
12
(Critical value)
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Example Metro EMS

• One-Tailed Test about a Population Mean ? Known
• Let n 40, 13.25 minutes, ?
  3.2 minutes
• Since 2.47 gt 1.645, we reject H0.
• Conclusion We are 95 confident that Metro
  EMS
• is not meeting the response goal of 12
  minutes
• appropriate action should be taken to improve
• service.

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Example Metro EMS

• Using the p-value to Test the Hypothesis
• Recall that z 2.47 for 13.25. Then
  p-value .0068.
• Since p-value lt ?, that is .0068 lt .05, we
  reject H0.

Reject H0
Do Not Reject H0
p-value???????
z
0
1.645
2.47
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Example Metro EMS

• 95 one-sided confidence interval calculation
• We are 95 confident that the mean is at least as
  large as 12.417
• We are 95 confident that Metro EMS is not
  meeting the response goal of 12 minutes.
  Therefore Reject The Null Hypothesis

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Two-Tailed Tests about a Population Mean s
Known

• Hypotheses
• H0 ????? ?
• Ha? ???????
• Test Statistic ? ?Known
• Rejection Rule Reject H0 if z gt z???

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Example Glow Toothpaste

• Two-Tailed Tests about a Population Mean The
  production line for Glow toothpaste is designed
  to fill tubes of toothpaste with a mean weight of
  6 ounces.
• 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 ounces otherwise the
  filling process will be stopped and adjusted.

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Example Glow Toothpaste

• Two-Tailed Tests about a Population Mean Large
  n
• A hypothesis test about the population mean can
  be used to help determine when the filling
  process should continue operating and when it
  should be stopped and corrected.
• Hypotheses
• H0 ????? ?
• ??????Ha? ??????
• Rejection Rule
• ???????ssuming a .05 level of significance,
• Reject H0 if z lt -1.96 or if z gt 1.96

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Example Glow Toothpaste

• Two-Tailed Test about a Population Mean Large n

Sampling distribution of (assuming H0 is
true and ? 6)
Reject H0
Do Not Reject H0
Reject H0
??????????
??????????
z
0
1.96
-1.96
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First Method Glow Toothpaste

• Two-Tailed Test about a Population Mean Large n
• Assume that a sample of 30 toothpaste tubes
• provides a sample mean of 6.1 ounces. Standard
  deviation (population) of 0.2 ounces.
• Let n 30, 6.1 ounces, ? .2 ounces
• Since 2.74 gt 1.96, we reject H0.
• Conclusion We are 95 confident that the mean
  filling weight of the toothpaste tubes is not 6
  ounces. The filling process should be stopped
  and the filling mechanism adjusted.

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Second Method Glow Toothpaste

• Using the p-Value for a Two-Tailed Hypothesis
  Test
• Suppose we define the p-value for a two-tailed
  test as double the area found in the tail of the
  distribution.
• With z 2.74, the standard normal probability
• table shows there is a .5000 - .4969 .0031
  probability
• of a difference larger than .1 in the upper tail
  of the
• distribution.
• Considering the same probability of a larger
  difference in the lower tail of the distribution,
  we have
• p-value 2(.0031) .0062
• The p-value .0062 is less than ? .05, so H0 is
  rejected.

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Two-Tailed Tests About a Population Mean

• p-Value Approach

1/2 p -value .0031
1/2 p -value .0031
a/2 .025
a/2 .025
z
0
z 2.74
z -2.74
za/2 1.96
-za/2 -1.96
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Third MethodConfidence Interval Approach to
aTwo-Tailed Test about a Population Mean

• Select a simple random sample from the population
  and use the value of the sample mean to
  develop the confidence interval for the
  population mean ?.
• If the confidence interval contains the
  hypothesized value ?0, do not reject H0.
  Otherwise, reject H0.

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Example Glow Toothpaste

• Confidence Interval Approach to a Two-Tailed
  Hypothesis Test
• The 95 confidence interval for ? is
• or 6.0284 to 6.1716
• Since the hypothesized value for the population
  mean, ?0 6, is not in this interval, the
  hypothesis-testing conclusion is that the null
  hypothesis,
• H0 ? 6, can be rejected.

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Tests about a Population Mean? Unknown

• Test Statistic ? ?Unknown
• This test statistic has a t distribution with n
  - 1 degrees of freedom.
• Rejection Rule
• One-Tailed Two-Tailed
• H0 ?????? Reject H0 if t gt t?
• H0 ?????? Reject H0 if t lt -t?
• H0 ?????? Reject H0 if t gt t??? U

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p -Values and the t Distribution

• The format of the t distribution table provided
  in most statistics textbooks does not have
  sufficient detail to determine the exact p-value
  for a hypothesis test.
• However, we can still use the t distribution
  table to identify a range for the p-value.
• An advantage of computer software packages is
  that the computer output will provide the p-value
  for the
• t distribution.

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Example Highway Patrol

• One-Tailed Test about a Population Mean Small n
• A State Highway Patrol periodically samples
  vehicle speeds at various locations on a
  particular roadway. The sample of vehicle speeds
  is used to test the hypothesis
• H0 m lt 65.
• The locations where H0 is rejected are deemed
  the best locations for radar traps.
• At Location F, a sample of 16 vehicles shows a
  mean speed of 68.2 mph with a standard deviation
  of 3.8 mph. Use an a .05 to test the
  hypothesis.

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Example Highway Patrol

• One-Tailed Test about a Population Mean Small n
• Let n 16, 68.2 mph, s 3.8 mph
• a .05, d.f. 16-1 15, ta 1.753
• Since 3.37 gt 1.753, we reject H0.
• Conclusion We are 95 confident that the mean
  speed of vehicles at Location F is greater than
  65 mph. Location F is a good candidate for a
  radar trap.

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Excel

• Give below is a sample of monthly rent values ()
  for one-bedroom apartments. The data is a sample
  of 70 apartments in a particular city. The data
  are presented in ascending order.
• Lets suppose we want to test

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Continued

• The first step is to find
• Variable Name
• Number of Observations
• Lowest Value
• Mean
• Median
• Standard Deviation
• Standard Error
• Maximum Value
• 1st Quartile
• 3rd Quartile.

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Continued

• Variable Name Rent
• Number of Observations 70
• Lowest Value 425
• Mean 490.8
• Median 475
• Standard Deviation 54.74
• Standard Error 6.54
• Maximum Value 615
• 1st Quartile 446.25
• 3rd Quartile 552.5

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Null Hypothesis ?500
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Null Hypothesis ??515
Null Hypothesis ?? 515
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Null Hypothesis ??480
Null Hypothesis ?? 480
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Summary of Test Statistics to be Used in
aHypothesis Test about a Population Mean
Yes
No
n gt 30 ?
No
Popul. approx. normal ?
? known ?
Yes
Yes
Use s to estimate s
No
? known ?
No
Use s to estimate s
Yes
Increase n to gt 30
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A Summary of Forms for Null and Alternative
Hypotheses About a Population Proportion

• The equality part of the hypotheses always
  appears
• in the null hypothesis.

• In general, a hypothesis test about the value
  of a
• population proportion p must take one of
  the
• following three forms (where p0 is the
  hypothesized
• value of the population proportion).

One-tailed (lower tail)
One-tailed (upper tail)
Two-tailed
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Tests About a Population Proportion

• Test Statistic

where
assuming np gt 5 and n(1 p) gt 5
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Tests About a Population Proportion

• Rejection Rule p Value Approach

Reject H0 if p value lt a

• Rejection Rule Critical Value Approach

H0 p???p?
Reject H0 if z gt z?
H0 p???p?
Reject H0 if z lt -z?
H0 p???p?
Reject H0 if z lt -z??? or z gt z???
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Example NSC

• Two-Tailed Test about a Population Proportion
  Large n
• For a Christmas and New Years week, the
  National Safety Council estimated that 500 people
  would be killed and 25,000 injured on the
  nations roads. The NSC claimed that 50 of the
  accidents would be caused by drunk driving.
• A sample of 120 accidents showed that 67 were
  caused by drunk driving. Use these data to test
  the NSCs claim with a 0.05.

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Two-Tailed Test About a Population Proportion

• Critical Value Approaches

1. Determine the hypotheses.
2. Specify the level of significance.
a .05
3. Compute the value of the test statistic.
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Two-Tailed Test About a Population Proportion

• Critical Value Approach

4. Determine the criticals value and rejection
rule.
For a/2 .05/2 .025, z.025 1.96
Reject H0 if z lt -1.96 or z gt 1.96
5. Determine whether to reject H0.
Because 1.278 gt -1.96 and lt 1.96, we cannot
reject H0.
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End Lesson 3