More about Hypothesis Testing

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Chapter 15

• More about Hypothesis Testing

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More about Hypothesis Testing

• Hypothesis Testing An Overview
• Why hypothesis test?
• If we can measure every member of the population,
  do not need hypothesis test.
• The descriptive statistic of the entire
  population is what we are concerned about.
• If using a sample to estimate a population, need
  a clean method to determine when a difference
  between the ? and the suspected m indicates the
  hypothesis is wrong.
• Determine what the common and rare sample means
  are according to the hypothesis

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More about Hypothesis Testing

• Hypothesis Testing An Overview
• Significant Difference
• Sample means rarely equal m
• There is variability amongst the samples.
• Common samples have small distances between ? and
  mhyp
• No big deal! Difference due to random
  variability.
• No reason to suspect that the sample does not
  come from the hypothesized population
• Rare samples have large differences between
• Cause for concern.
• Signifies that the sample did not come from the
  hypothesized population.

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More about Hypothesis Testing
Sampling Distribution Significant Differences
m? mhyp
Small Dif
Large dif
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More about Hypothesis Testing

• Hypothesis Testing An Overview
• Importance of the Standard Error
• Significance is based upon a relative difference
• Is the difference between ? and mhyp greater than
  the typical sample from a population with m
  mhyp
• The standard error gives the typical distance
  from m?
• It is the standard deviation of the sampling
  distribution.

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More about Hypothesis Testing
Standard Error and Significant Differences
Large Standard Error
Small Standard Error
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More about Hypothesis Testing

• Hypothesis Testing An Overview
• The Hypothetical Sampling Distribution vs the
  Real Sampling Distribution
• Hypothetical Sampling Distribution
• Based upon mhyp
• Distribution of all possible samples according to
  H0
• Used to make decision rule
• Real Sampling Distribution
• Based upon the actual population mean (mreal)
• Distribution of all possible samples that can
  actually be selected.

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More about Hypothesis Testing

• Hypothesis Testing An Overview
• The Hypothetical Sampling Distribution vs the
  Real Sampling Distribution
• Goal of the Hypothesis Test
• Determine if the real sampling distribution is
  the same as the hypothetical sampling
  distribution.
• Determine if mreal mhyp

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More about Hypothesis Testing
Hypothetical Sampling Distribution
Potential Real Sampling Distributions
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More about Hypothesis Testing

• Hypothesis Testing An Overview
• Never prove anything! Only decide!
• Hypothetical populations go on forever (so do
  sampling distributions)
• Significant difference?
• May be because sample comes from a different
  population than hypothesized.
• May just be the odd ball sample.

m?
Although these samples are extremely unlikely,
they are still possible
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More about Hypothesis Testing

• Hypothesis Testing An Overview
• Possibility of an Incorrect Decision
• Might get oddball sample
• Hypothesis test allows us to discuss the
  probability of making an error
• Minimizing incorrect decisions
• Incorrect decisions are only a problem if mhyp is
  really close, but not equal, to mreal

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More about Hypothesis Testing
Hypothetical Sampling Distribution
Potential Real Sampling Distributions
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More about Hypothesis Testing

• Strong or Weak Decisions
• Retaining H0 is a weak decision.
• Decision is that sample is common.
• Might also be a common sample in a slightly
  different sampling distribution.
• mhyp ? mreal
• So, unsure if H0 is correct, or just close to
• Can not place a value on the likelihood of an
  incorrect decision

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More about Hypothesis Testing

• Strong or Weak Decisions
• Rejecting H0 is a strong decision
• Decision is that the sample is rare.
• Probably not from the hypothetical sampling
  distribution.
• Indisputable! Even if unsure what mreal actually
  is.
• Can place a value on the likelihood of an
  incorrect decision

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More about Hypothesis Testing
Hypothetical Sampling Distribution
Potential Real Sampling Distributions
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More about Hypothesis Testing

• Why the Research Hypothesis is not directly
  tested!
• Concern of the researcher.
• Usually, the null hypothesis is the secondary
  concern.
• The primary concern is the alternative
  hypothesis!
• Is the height of UB students different than the
  nation?
• Does tequila make people more drunk?
• Are the convention members animals in disguise?

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More about Hypothesis Testing

• Why the Research Hypothesis is not directly
  tested!
• Lacks necessary precision.
• We need a in order to draw hypothetical sampling
  distribution and decision rule.
• Alternative hypothesis usually does not specify a
  value.
• The height of UB students is different than the
  nation.
• What do you think it is?
• Tequila makes people more drunk.
• How much more drunk do you think they get?

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More about Hypothesis Testing

• Why the Research Hypothesis is not directly
  tested!
• Deciding to conclude with the alternative
  hypothesis is a strong decision.
• Rejecting the null hypothesis is a strong
  decision.

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One-tailed Tests

• Only difference is the form of the hypotheses and
  the shading to determine the critical values

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One-Tailed Tests

• Every hypothesis test we have done so far has
  been a two-tailed test (a.k.a., a non-directional
  test).
• H1 does not specify the direction that mreal
  might be
• If decide to reject H0
• Rejection region split between two tails
• a cut in half and shaded at both ends

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One-Tailed Tests

• Two Tailed Tests
• Only two potential interpretations (technically
  speaking)
• Retain H0 m 68
• We have no evidence to indicate that the
  population mean is 68.
• Reject H0go with H0 m ? 68
• The population mean is not 68.

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One-Tailed Tests

• One-Tailed (Directional) Tests
• What if we..
• wanted to make a conclusion about the direction
  of mreal.
• had reasons to believe that mreal is only
  above/below H0
• From Outside Research
• Toxins only stunt growth.
• We need a(n)
• H1 m lt 68
• test that is only concerned about rejecting H0
  if ? is significantly less than mhyp.

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One-Tailed Tests

• One-Tailed (Directional) Tests
• Alternative Hypothesis states a direction
• H1 m lt 68
• H1 m gt 68
• Null Hypothesis incorporates opposite direction
  of H1 (still has equals).
• H0 m ? 68
• H0 m ? 68

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One-Tailed Tests

• One-Tailed (Directional) Tests
• Shade entire rejection region (a) in tail
  indicated by H1
• H1 m lt 68
• Shade lower tail
• H1 m gt 68
• Shade upper tail

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One-Tailed Tests
One-tailed Lower-Critical Test
H0 m 68
Step 2)
H1 m lt 68
lt mhyp mhyp gt mhyp
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One-Tailed Tests
One-tailed Lower-Critical Test

• Hypothesis
• H0 m ? 68
• H1 m lt 68

Step 3)
zcrit -1.65
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One-Tailed Tests
One-tailed Upper-Critical Test
H0 m 68
Step 2)
H1 m gt 68
lt mhyp mhyp gt mhyp
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One-Tailed Tests
One-tailed Lower-Critical Test

• Hypothesis
• H0 m ? 68
• H1 m gt 68

Step 3)
zcrit 1.65
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Hypothesis Test 5

• Little Red Ridinghood uses a 1-tail
  Lower-critical Test

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Little Red Ridinghood
Little Red Ridinghood is catering cookies for
the Inaugural Blue Valley Grandmother Convention.
But after her experience with the wolf, she has
been really paranoid about the possibility that a
Grandmother may really be some animal that wants
to eat her. So she developed the Grandmother
Aptitude Test (GAT). On the GAT, grandmothers
have a population mean of 1000 and a standard
deviation of 200. Red wants to use the GAT to
test the hypothesis that this convention really
is a group of grandmothers, as opposed to
something else (at the .05 level of
significance). Lil Red is only concerned about
a population that might have a lower mean on the
GAT. If they have a higher mean, they are simply
extraordinary grandmothers. To perform this
test, Red randomly gave the GAT to 100 convention
members, incognito. She got a sample mean of 965.
Help Red perform this hypothesis test.
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Little Red Ridinghood
Little Red Ridinghood is catering cookies for
the Inaugural Blue Valley Grandmother Convention.
But after her experience with the wolf, she has
been really paranoid about the possibility that a
Grandmother may really be some animal that wants
to eat her. So she developed the Grandmother
Aptitude Test (GAT). On the GAT, grandmothers
have a population mean of 1000 and a standard
deviation of 200. Red wants to use the GAT to
test the hypothesis that this convention really
is a group of grandmothers, as opposed to
something else (at the .05 level of
significance). Lil Red is only concerned about
a population that might have a lower mean on the
GAT. If they have a higher mean, they are simply
extraordinary grandmothers. To perform this
test, Red randomly gave the GAT to 100 convention
members, incognito. She got a sample mean of 965.
Help Red perform this hypothesis test.
mGMs1000 sGMs200 mhyp-Conv1000 sConv200 n
100 ? 965 a .05 1-Tail Lower
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Little Red Ridinghood

• Step 1) Rewrite Research Question
• Is the mean of the convention members 1000?

mhyp-Conv1000 sConv200 n 100 ? 965 a .05
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Little Red Ridinghood
Step 2) Write the statistical hypotheses
H0 m 68
H1 m lt 1000
lt mhyp mhyp gt mhyp
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Little Red Ridinghood

• Step 3) Form Decision Rule
• Draw Normal Curve
• Shade in a
• Mark Rejection Region(s)
• Determine Critical Scores
• Write conditions for rejection H0

• Hypothesis
• H0 mconv ? 1000
• H1 mconvlt 1000

zcrit -1.65
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Little Red Ridinghood
m? m
mhyp
1000

• Hypothesis
• H0 mconv ? 1000
• H1 mconvlt 1000

s?

• Decision Rule
• Reject H0
• zobt lt -1.65

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Little Red Ridinghood
Step 4) Calculate Test Statistic
zobt (? - mhyp) / s?
zobt (965 - 1000) / 20
zobt - 35 / 20

• Hypothesis
• H0 mconv ? 1000
• H1 mconvlt 1000

zobt -1.75

• Decision Rule
• Reject H0
• zobt lt -1.65

Based upon a sampling distribution with m?
1000 s? 20
.3
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Little Red Ridinghood
Step 5) Make Decision
Step 6) Interpret Decision

• The convention members are not grandmothers.
  Their mean on the GAT is less than 1000

• Hypothesis
• H0 mconv ? 1000
• H1 mconvlt 1000

• Hypothesis
• H0 mconv ? 1000
• H1 mconvlt 1000

• Decision Rule
• Reject H0
• zobt lt -1.65

Test Statistic zobt -1.75
Based upon a sampling distribution with m?
1000 s? 20
.3
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One-tailed Test

• Additional Notes

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One-tailed Tests

• One-Tailed tests are more sensitive in the
  appropriate direction.
• Smaller differences cause them to reject H0
• Smaller minimum significant differences
• zcrit closer to 0

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One-tailed Tests

• Only use one-tailed tests if you have a valid
  reason for believing mreal is in a specific
  direction.
• Weakens strength of decision to reject H0

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One-tailed Tests

• Only use one-tailed tests if you have a valid
  reason for believing mreal is in a specific
  direction.
• Impossible to reject H0 if mreal is in the other
  direction
• Regardless of how wrong H0 is.

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One-tailed Tests

• If unstated, use a 2-tail test.
• If reject on a 2-tail, will reject on a 1 tail in
  that direction.

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Level of Significance

• Choosing Something Else

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Level of Significance

• Can it be something other than .05?
• Yes! Alpha can be set to whatever you need.
• Why would I want to change alpha?
• If the ramifications for rejecting H0 are costly,
  might want to use a .01.
• Less sensitive Harder to Reject
• Larger minimum significant differences
• zcrit farther from 0

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Level of Significance

• How does changing alpha change a hypothesis test.
• Only changes Step 3)
• Shade a total of .01
• Two-tailed .005 in each tail
• One-tailed .01 in appropriate tail

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Hypothesis Test 6

• Example Height of UB Students with a .01

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UB Height Example
It has long been known that the national mean
for height is 58 (68) with a standard
deviation of 6. Typically, it has been assumed
that UB students should be equal to the national
average. However, a researcher became concerned
with the average height of UB students after
discovering the enormous amount of toxins in the
ground in the WNY region. If the researcher
determines that the height of UB students is not
equal to the national average, the NYS government
will enact a 3 billion clean-up and research
effort on the problem. In order to not waste
money, the government has asked the research to
be very certain if there is a difference.
Therefore, the researcher will adopt a .01 level
of significance To test the hypothesis, the
research measured the height of 100 randomly
selected UB students. The mean height of this
sample was 59.5 (69.5). What will this
researcher conclude about the height of UB
Students?
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UB Height Example
It has long been known that the national mean
for height is 58 (68) with a standard
deviation of 6. Typically, it has been assumed
that UB students should be equal to the national
average. However, a researcher became concerned
with the average height of UB students after
discovering the enormous amount of toxins in the
ground in the WNY region. If the researcher
determines that the height of UB students is not
equal to the national average, the NYS government
will enact a 3 billion clean-up and research
effort on the problem. In order to not waste
money, the government has asked the research to
be very certain if there is a difference.
Therefore, the researcher will adopt a .01 level
of significance To test the hypothesis, the
research measured the height of 100 randomly
selected UB students. The mean height of this
sample was 59.5 (69.5). What will this
researcher conclude about the height of UB
Students?
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UB Height Example

• Step 1) Rewrite Research Question
• Is the mean height of UB students 68?

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UB Height Example
Is the mean height of UB students 68?

• Step 2) Write the statistical hypotheses
• Null Hypothesis
• The mean height of UB Students is 68
• H0 mUB 68
• Alternative Hypothesis
• The mean height of UB Students is not 68
• H1 mUB ? 68

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UB Height Example
Is the mean height of UB students 68?

• Step 3) Form Decision Rule
• Draw Normal Curve
• Shade in a
• Mark Rejection Region(s)
• Determine Critical Scores
• Write conditions for rejection H0

• Hypothesis
• H0 mUB 68
• H1 mUB ? 68

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UB Height Example
Is the mean height of UB students 68?
m? m
mhyp
68

• Hypothesis
• H0 mUB 68
• H1 mUB ? 68

s?

• Decision Rule
• Reject H0
• zobt ? -2.58
• zobt ? 2.58

.6
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UB Height Example
Is the mean height of UB students 68?
Step 4) Calculate Test Statistic
zobt (? - mhyp) / s?
zobt (69.5 - 68) / .6
zobt 1.5 / .6

• Hypothesis
• H0 mUB 68
• H1 mUB ? 68

zobt 2.5

• Decision Rule
• Reject H0
• zobt ? -2.58
• zobt ? 2.58

Based upon a sampling distribution with m? 68
s? .6
.3
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UB Height Example
Is the mean height of UB students 68?
Step 5) Make Decision
Step 6) Interpret Decision

• There is no evidence to conclude that the mean
  height of UB students does not equal the national
  average.

• Hypothesis
• H0 mUB 68
• H1 mUB ? 68

• Decision Rule
• Reject H0
• zobt ? -2.58
• zobt ? 2.58

Test Statistic zobt 2.5
Based upon a sampling distribution with m? 68
s? .3
.3