Week 10

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Week 10
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OBJECTIVES

• TO DESCRIBE THE MAJOR CHARACTERISTICS OF
  STUDENTS t- DISTRIBUTION.
• TO UNDERSTAND THE DIFFERENCE BETWEEN THE t
  -DISTRIBUTION AND THE z -DISTRIBUTION.
• TO CONSTRUCT CI FOR THE MEAN OF A POPULATION, ?
• TO TEST A HYPOTHESIS INVOLVING ONE POPULATION
  MEAN

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Inference for the Mean of a Population
So far we have based our analysis on unrealistic
assumption that the population standard deviation
is known.
ASSUMPTIONS CONDITIONS (1) Observations in
the sample are INDEPENDENT (2) Population has a
NORMAL distribution
OR The sample is LARGE (n gt 30) (3)
Standard deviation of the POPULATION. s, is
UNKNOWN.
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Characteristics of the Studentst-distribution
The t-distribution has the following
properties 1- It is continuous, bell shaped and
symmetrical about zero like the
z-distribution. 2- There is a family of
t-distributions with mean of zero but different
standard deviations. 3- The t-distribution is
more spread out and flatter at the center than
the z-distribution, but approaches the
z-distribution as the sample size gets larger.
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Concept of Degrees of Freedom
Refers to freedom of observations to vary after
some RESTRICTIONS have been imposed.
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THE ONE-SAMPLE t PROCEDURES a) Confidence
Interval for the m
where t is the critical value for the t(n-1)
distribution and s SAMPLE standard deviation
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THE ONE-SAMPLE t PROCEDURES b) The
one-sample t-test
has a t distribution with n-1 degrees of freedom
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Calculation of the P-value
Ha ? gt ?o
P( T gt t )
Ha ? lt ?o
to find the P-value use
P( T lt t )
Ha ? ? ?o
2P( T gt t )
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• Robustness
• A confidence interval or Hypothesis test is said
  to be robust if the confidence level or p-value
  does not change very much when the assumptions of
  the procedure are violated.
• Using the t Procedures
• Except in small samples, the assumption of a SRS
  from the population is more important than the
  assumption that the population distribution is
  normal.
• Sample size is less than 30
• Use t procedures if the data are close to
  normal
• Sample size is large (n gt 30)
• t procedures can be used even for clearly
  skewed distributions

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EXAMPLE 1
A pharmacy student is hopelessly enslaved by the
smoking habit. Although she is aware that she
may contract lung cancer (fast taking over from
breast cancer as the biggest killer of Australian
women - more than 25 per 1,000 women) she finds
it hard to quit. Suppose that it has been
established that if cigarettes average 30
milligrams or more of nicotine, lung cancer is
certain to develop in the user. The smoker is
willing to take her chances if her brand has an
average less than 30 mg. (Assume that the
population is normal). A random sample of 25 of
her brand of cigarettes yield a mean nicotine
level of 29 mg with the standard deviation 2.63
mg. What decision is the smoker likely to take?
a) Use a 0.01. Show all working. b) What is
the decision if the significance level is ?
0.05?
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EXAMPLE 1
c) What would be your decision if you were given
the following NORMAL PROBABILITY PLOT of the
sample data.
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EXERCISE 1
The one sample t statistic from a sample of n
20 observations for the two sided test
of H0 ? 120 Ha ? ? 120 has the
value 1.8. a) What are the degrees of freedom for
t? b) Locate the two critical values t from
table D that bracket t. What are the right tail
probabilities for these two values? c) How would
you report the P-value for this test? d) Is the
value t 1.8 statistically significant at the 5
level?
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Using Minitab to conduct one-sample t procedures
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Using Minitab to conduct one-sample t procedures
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Using Minitab to conduct one-sample t procedures
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COMPARING TWO POPULATION MEANSO B J E C T I V E S

• A) TO CONDUCT A TEST OF HYPOTHESIS FOR THE
  DIFFERENCE BETWEEN A SET OF PAIRED OBSERVATIONS.
• B) TO CONSTRUCT CI AND TEST A HYPOTHESIS
  INVOLVING THE DIFFERENCE BETWEEN THE MEANS OF TWO
  INDEPENDENT POPULATIONS (groups).

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A) HYPOTHESIS TESTING INVOLVING PAIRED
OBSERVATIONS

• Use this procedure when TWO samples are
  dependent.
• There are two common experimental designs where
  this procedure is applicable
• 1. Subjects are matched in pairs and each
  treatment is given to one subject in each pair
• 2. Before and after observations on the same
  subjects
• This reduces to the one-sample t procedures to
  the observed DIFFERENCES
• ASSUMPTION Population of differences is NORMAL

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EXAMPLE 2
In a small clinical trial to assess the value of
a new tranquilizer on psychoneurotic patients,
each patient was given a weeks treatment with a
drug and a weeks treatment with a placebo, in
random order. At the end of each weak the patient
had to complete a questionnaire, on the basis of
which they were given an anxiety score (with
possible value from 0 to 30), high scores
corresponding to states of anxiety. The results
are shown in Table 1. Assuming that the
population of differences follows a normal
distribution, at the 0.05 level test the
hypothesis that the new drug reduce anxiety!
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Table 1 Anxiety scores recorded for 10 patients
receiving a new drug and a placebo in random order
Patient Drug Placebo Difference (drug
- placebo) 1 19 22 -3 2 11 18 -7 3 14
17 -3 4 17 19 -2 5 23 22
1 6 11 12 -1 7 15 14 1 8 19 11
8 9 11 19 -8 10 8 7 1
-13
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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
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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
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Using Minitab to test H0
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Using Minitab to test H0
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Minitab output for paired data
MTB gt Paired 'Drug' 'Placebo' SUBCgt
Confidence 95.0 SUBCgt Test 0.0 SUBCgt
Alternative -1. Paired T-Test and Confidence
Interval Paired T for Drug - Placebo
N Mean StDev SE Mean Drug
10 14.80 4.69 1.48 Placebo
10 16.10 4.95 1.57 Difference
10 -1.30 4.55 1.44 95 CI
for mean difference (-4.55, 1.95) T-Test of mean
difference 0 (vs lt 0) T-Value -0.90 P-Value
0.195
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B) COMPARING POPULATION MEANSOF TWO
INDEPENDENT POPULATIONS

• Two Sample Problem
• The goal is to compare the responses of two
  treatments or to compare the characteristics of
  two populations.
• We have a separate sample from each treatment or
  each population.
• Assumptions
• (i) A SRS is taken independently, one from each
  of the two distinct populations.
• (ii) Both populations are normally distributed. .

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Two independent samples
Samples come from 2 INDEPENDENT
populations. There is NO MATCHING of the elements
in the two sample. Two samples can be of
different sizes.
Sample 1
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Two independent samples
1. Two-sample Z-test and Z confidence Interval
2. Two-sample
t-test and t Confidence Interval 3. If
we impose an extra assumption, that the both
populations have equal variances, we can apply
Pooled two-sample t-test
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Two independent samples
All these procedures have the same generic forms
Confidence Intervals
Test Statistics
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• 1. Two-sample Z statistic
• Parameters for the two populations are m1,
  m2, s1, and s2.
• EXTRA (unrealistic) ASSUMPTION s1 AND s2 are
  known
• The test statistic is

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2. Two-sample t procedures
NO EXTRA ASSUMPTIONS
I) To test the hypothesis H0 ?1 ?2
Standard Error
II) To form the confidence interval for ?1 - ?2

• Drawback The two-sample t statistic does not
  have exact t distribution.
• Number of degrees of freedom is equal to
  (smaller sample size - 1)

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3. POOLED Two-sample t-test
ADDITIONAL ASSUMPTION The population VARIANCES
must be EQUAL
Number of Degrees Of Freedom n1
n2 - 2
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Two-sample tests REVIEW
ADDITIONAL ASSUMPTION Population VARIANCES are
KNOWN
NO ADDITIONAL ASSUMPTIONS
ADDITIONAL ASSUMPTION The population VARIANCES
must be EQUAL.
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Two-sample t-Confidence Intervals

• The POOLED two-sample t Confidence Interval

• Number of degrees of freedom and assumptions are
  the SAME as for the corresponding tests!

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EXAMPLE 3 - 1998 exam question
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EXAMPLE 3 - 1998 exam question
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EXAMPLE 3 - 1998 exam question
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• The logic of hypothesis tests is not intuitively
  obvious. However, it is parallel to that used in
  the courtroom trials to decide whether a
  defendant is innocent or guilty of a crime.
• Ho The defendant is innocent Ha The
  defendant is guilty
• In reaching the verdict, the defendant is
  presumed innocent, and it is concluded that he or
  she is guilty only when the evidence establishes
  this beyond reasonable doubt. We are never 100
  sure that the defendant is either innocent or
  guilty.
• As in the courtroom, the Ho has a special status
  In carrying out a test we will presume that it is
  true we need an overwhelming evidence to show
  that it is false.
• The same is with significance tests we do not
  draw conclusions about which we are 100 certain.
  Why? Because the decision is made on the SAMPLE
  evidence.

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• Although we ALWAYS state our research statement
  as an alternative hypothesis, it is the null
  hypothesis that plays a central role in testing
  because
• the test statistic measures how far the observed
  data departs from what is expected under the null
  hypothesis
• the P-Values is calculated assuming that the null
  hypothesis is true

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Type I and Type II Errors
Type I Error Rejecting the null hypothesis, H0,
when it is actually true. Type II Error Failing
to reject the null hypothesis, H0, when it is
actually false.
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JURYS DILEMMA
TRUTH
INNOCENT
GUILTY
Correct (innocent cleared)
TYPE 2 (?) ERROR (gulity cleared)
ACQUIT
DECISION
Correct (guilty convicted)
TYPE 1 (?) ERROR (innocent convicted)
CONVICT
Ho The defendant is innocent Ha The defendant
is guilty
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Level of significance ? and the power of a test
Level of Significance The probability of
rejecting the null hypothesis when it is
actually true, i.,e., value of ?. It specifies
probability that will be classified as unlikely
enough to consider that the evidence against the
null hypothesis is overwhelming. It is the level
of risk we are prepared to take in making a type
I error. ? 0.05 ? Anything that occurs less
than 1 in 20 times is said to be unlikely. In
other words, we are prepared to accept a 1 in 20
chance that we have rejected the null hypothesis
when it is true. Power of a test 1 - ?, is
ability of the test to reject null hypothesis
when it is false.
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Is it better to make an ? error or a ? error? 3
safeguards in the jury system are designed to
make the error ? small a) Defendant is presumed
to be innocent until proved guilty. b) Jury must
be convinced beyond reasonable doubt. c) Every
member of the jury must be convinced of the
defendants guilty.