Week 9a

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Week 9a
HYPOTHESIS TESTING
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O U T L I N E
1. The Nature of Hypothesis Testing 2 . One and
Two-tailed Statistical tests 3. A six-step
Model 4. Hypothesis Test of a Mean (? known)
A Probability-Value Approach 5. Hypothesis
Test of a Mean (? known) A Classical
Approach
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Statistical Inference

• IF NOTHING IS KNOWN ABOUT THE POPULATION,
  ESTIMATION IS USED TO PROVIDE POINT AND INTERVAL
  ESTIMATION ABOUT THE POPULATION.
• IF INFORMATION ABOUT THE POPULATION IS CLAIMED OR
  SUSPECTED, HYPOTHESIS TESTING IS USED TO
  DETERMINE THE FEASIBILITY OF THIS INFORMATION.
  

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WHAT IS HYPOTHESIS TESTING?
Hypothesis A statement about the value of a
population parameter stated for the purpose of
testing. Hypothesis testing A procedure, based
on sample evidence and probability theory, used
to determine whether the hypothesis is a
reasonable statement and should not be rejected,
or is unreasonable and should be rejected. Does
the sample result reflect a real change?
Or Could we as easily
get the same outcome by chance?
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A Six-Step Hypothesis Test Model
State the null and the alternative hypotheses
and nominate the significance level, ?
STEP 1
STEP 2
Decide which test to use and obtain test statistic
Check the assumptions and conditions
STEP 3
STEP 4
Obtain the p-value (or determine critical
value(s))
Formulate and apply a decision rule
STEP 5
State the conclusion
STEP 6
Reject H0 and accept H1
Do not reject H0
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STEP 1 Set up the Hypotheses and
Nominate Level of Significance

• Null Hypothesis H0 A statement about the value
  of a population parameter. It denotes the status
  quo of the parameter in the population (No
  effect or No difference).
• This is a claim that we will try to find evidence
  AGAINST.
• The goal of testing is to assess the strenght of
  the evidence against the null hypothesis.
• Alternative Hypothesis Ha A statement that is
  accepted if the sample data provide evidence that
  the null hypothesis is false.
• We arbitrarily decide on how much evidence
  against H0 will be regarded as decisive.
• That probability is called the level of
  significance, and is denoted by ?. It is common
  to specify ? 0.05 or 0.01.

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STEP 2 Select APPROPRIATE test and obtain a test
statistic
Take a sample and calculate the appropriate test
statistic. A test statistic is a quantity
calculated from sample data to assist us in
making a decision Its value determines whether
we reject or do not reject Ho
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STEP 3 Check the assumptions and conditions

• 1. Observations are independent
• (this is always the case when the sample is
  random)
• TO APPLY A PARAMETRIC TEST (Z or t) ONE
  OF THE
• FOLLOWING CONDITIONS MUST BE MET
• 2. THE POPULATION HAS A NORMAL DISTRIBUTION
• (This is called a NORMALITY
  assumption)
• O
  R
• THE SAMPLE IS LARGE (n gt 30) and we can
  apply the CLT
• 3. Z-test is used when the standard deviation of
  the population is KNOWN
• t-test when ? is unknown (almost ALWAYS the
  case in practice)

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Parametric vs Nonparametric Tests

• Parametric Tests
• Validity of these test depends on a rather
  strict set of assumptions concerning population
  distribution
• 1. Normality
• Later we will introduce parametric tests that
  require an additional assumption
• Variances of the populations are equal

• What are the Nonparametric Tests?
• Statistical Tests which do not require us to make
  assumptions about population distribution

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Nonparametric Statistics (Distribution Free
Statistics)

• Why do we need to know about Nonparametric
  statistics?
• Sometimes we collect data which is markedly
  skewed and to which we cannot VALIDLY apply the
  parametric tests
• Parametric tests assume a normal distribution
• t tests assume normal distribution
• z tests assume normal distribution
• When assumptions required by the parametric test
  are violated we will apply at test which does
  NOT assume a normal distribution

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STEP 4 Obtain the P-value (or determine
critical value(s))
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STEP 4 Obtain the P-value (or determine
critical value(s))
Critical value (or Critical z or t value) The
dividing point between the region where the null
hypothesis is rejected and the region where it is
not rejected. This value is found from the z or
the t table.
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STEP 4 Obtain the P-value (or determine
critical value(s))
Modern approach Calculate P-value for the
sample data. P-value is the probability,
calculated assuming that H0 is true, that the
test statistic would take a value as extreme or
more extreme than that actually observed.
Small P-values give evidence against H0. Large
P-values fail to reject Ho
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STEP 5 Modern approach
Formulate and apply the decision rule
IF the P-value ? the level of
significance ? THEN
Reject the null hypothesis and accept Ha
ELSE Fail to
reject the null hypothesis
RULE OF THUMB P-value ?
0.05 STATISTICALLY SIGNIFICANT The
result is not likely to be due to the chance
alone.
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Sampling Distribution for the Statistic Z for a
Two-Tailed Test, 0.05 Level of Significance
Do not reject H0
0.95
0.025
0.025
z
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Sampling Distribution for the Statistic Z for a
One-Tailed Test, 0.05 Level of Significance
0.95 Probability
0.05
z
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STEP 5 Classical approach
Formulate and apply the decision rule
IF the test statistic falls within the
critical region THEN
Reject the null hypothesis and accept Ha
ELSE Fail to
reject the null hypothesis
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EXERCISE 1
Which of the following statements is correct?
a. An extremely small p-value indicates that
the actual data differs markedly from
that expected if the null hypothesis were true.
b. The p-value measures the probability
that the hypothesis is true. c. The
p-value measures the probability of making a Type
II error. d. The larger the p-value, the
stronger the evidence against the null
hypothesis e. A large p-value indicates
that the data is consistent with the
alternative hypothesis.
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EXERCISE 2
In a hypothesis testing problem a. The null
hypothesis will not be rejected unless the data
are not unusual (given that the
hypothesis is true). b. The null hypothesis will
not be rejected unless the p-value indicates the
data are very unusual (given that the hypothesis
is true). c. The null hypothesis will not be
rejected only if the probability of
observing the data provide convincing evidence
that it is true. d. The null hypothesis is also
called the research hypothesis the
alternative hypothesis often represents the
status quo. e. The null hypothesis is the
hypothesis that we would like to prove the
alternative hypothesis is also called the
research hypothesis.
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Tests for a Population Mean - Example
Let us test a hypothesis that the average height
of Australians is 175 cm. Suppose that the
population has a normal distribution and that its
standard deviation is known, and equals 25. A
random sample of 25 persons has given average
height 172 cm. Is there enough evidence to reject
the null hypothesis?
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State the null and the alternative hypotheses
and nominate the significance level, ?
STEP 1

• H0 ? 175cm
• Ha ? ? 175cm (two tailed)
• OR
• Ha ? gt 175 (one tailed)
• OR
• Ha ? lt 175 (one tailed)
• ?.05

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Decide which test to use and obtain test statistic
STEP 2
Check the assumptions and conditions
STEP 3
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Obtain the p-value (or determine critical
value(s))
STEP 4
Ha ? gt ?o
P( Z gt z )
Ha ? lt ?o
to find the P-value use
P( Z lt z )
Ha ? ? ?o
2P( Z gt z )
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Formulate and apply a decision rule
STEP 5
State the conclusion
STEP 6
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EXAMPLE 2 A pharmaceutical manufacturer checks
the potency of products during manufacture by
chemical analysis. The standard release potency
for cephalotin crystals is set to 910. An assay
of the previous 5 lots gives the following
potency data 897 913 906
895 908 Is there significant evidence at the
5 level that the mean potency is less than the
standard release potency if it is known that the
population distribution is normal with standard
deviation equal 8?
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MTB gt Describe 'Potency'. Descriptive
Statistics Variable N Mean Median
TrMean StDev SE Mean Potency 5
903.80 906.00 903.80 7.60
3.40
MTB gt ZTest 910 8 'Potency' SUBCgt Alternative
-1. Z-Test Test of mu 910.00 vs mu lt
910.00 The assumed sigma 8.00 Variable N
Mean StDev SE Mean Z
P Potency 5 903.80 7.60 3.58
-1.73 0.042
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EXERCISE 3

• 1998 Exam question (1 mark)
• A significance test was performed to test the
  null hypothesis
• Ho ? 10 versus the alternative H1 ? gt 10.
• The test statistic is z 2.12. The P-value for
  this test is approximately
• (a) 0.983
• (b) .9830
• (c) 0.027
• (d) 0.017
• (e) 0.170