Parametric Statistical Inference

1
Parametric Statistical Inference
Instructor Ron S. Kenett Email
ron_at_kpa.co.il Course Website www.kpa.co.il/biosta
t Course textbook MODERN INDUSTRIAL
STATISTICS, Kenett and Zacks, Duxbury Press, 1998
2
Course Syllabus

• Understanding Variability
• Variability in Several Dimensions
• Basic Models of Probability
• Sampling for Estimation of Population Quantities
• Parametric Statistical Inference
• Computer Intensive Techniques
• Multiple Linear Regression
• Statistical Process Control
• Design of Experiments

3
Definitions

• Null Hypotheses
• H0 Put here what is typical of the population,
  a term that characterizes business as usual
  where nothing out of the ordinary occurs.
• Alternative Hypotheses
• H1 Put here what is the challenge, the view of
  some characteristic of the population that, if it
  were true, would trigger some new action, some
  change in procedures that had previously defined
  business as usual.

4
The Logic of Hypothesis Testing

• A new claim is asserted that challenges existing
  thoughts about a population characteristic.
• Suggestion Form the alternative hypothesis
  first, since it embodies the challenge.

• Step 1.
• A claim is made.

5
The Logic of Hypothesis Testing

• Step 2.
• How much error are you willing to accept?

• Select the maximum acceptable error, a. The
  decision maker must elect how much error he/she
  is willing to accept in making an inference about
  the population. The significance level of the
  test is the maximum probability that the null
  hypothesis will be rejected incorrectly, a Type I
  error.

6
The Logic of Hypothesis Testing

• Assume the null hypothesis is true. This is a
  very powerful statement. The test is always
  referenced to the null hypothesis.
• Form the rejection region, the areas in which
  the decision maker is willing to reject the
  presumption of the null hypothesis.

• Step 3.
• If the null hypothesis were true, what would you
  expect to see?

7
The Logic of Hypothesis Testing

• Step 4.
• What did you actually see?

• Compute the sample statistic. The sample
  provides a set of data that serves as a window to
  the population. The decision maker computes the
  sample statistic and calculates how far the
  sample statistic differs from the presumed
  distribution that is established by the null
  hypothesis.

8
The Logic of Hypothesis Testing

• Step 5.
• Make the decision.

• The decision is a conclusion supported by
  evidence. The decision maker will
• reject the null hypothesis if the sample evidence
  is so strong, the sample statistic so unlikely,
  that the decision maker is convinced H1 must be
  true.
• fail to reject the null hypothesis if the sample
  statistic falls in the nonrejection region. In
  this case, the decision maker is not concluding
  the null hypothesis is true, only that there is
  insufficient evidence to dispute it based on this
  sample.

9
The Logic of Hypothesis Testing

• Step 6.
• What are the implications of the decision for
  future actions?

• State what the decision means in terms of the
  research program.
• The decision maker must draw out the
  implications of the decision. Is there some
  action triggered, some change implied? What
  recommendations might be extended for future
  attempts to test similar hypotheses?

10
Two Types of Errors

• Type I Error
• Saying you reject H0 when it really is true.
• Rejecting a true H0.
• Type II Error
• Saying you do not reject H0 when it really is
  false.
• Failing to reject a false H0.

11
What are acceptable error levels?

• Decision makers frequently use a 5 significance
  level.
• Use a 0.05.
• An a-error means that we will decide to adjust
  the machine when it does not need adjustment.
• This means, in the case of the robot welder, if
  the machine is running properly, there is only a
  0.05 probability of our making the mistake of
  concluding that the robot requires adjustment
  when it really does not.

12
Three Types of Tests

• Nondirectional, two-tail test
• H1 pop parameter n.e. value
• Directional, right-tail test
• H1 pop parameter value
• Directional, left-tail test
• H1 pop parameter
• Always put hypotheses in terms of population
  parameters and have
• H0 pop parameter value

13
Two tailed test
H0 pop parameter value H1 pop parameter n.e.
value
14
Right tailed test
H0 pop parameter value H1 pop parameter
value
15
Left tailed test
H0 pop parameter value H1 pop parameter value
16
Reality
Ho
H1
Type I Error
H1
OK
Decision
Type II Error
OK
Ho
17
What Test to Apply?

• Ask the following questions
• Are the data the result of a measurement (a
  continuous variable) or a count (a discrete
  variable)?
• Is s known?
• What shape is the distribution of the population
  parameter?
• What is the sample size?

18
Test of µ, s Known, Population Normally
Distributed

• Test Statistic
• where
• is the sample statistic.
• µ0 is the value identified in the null
  hypothesis.
• s is known.
• n is the sample size.

19
Test of µ, s Known, Population Not Normally
Distributed

• If n 30, Test Statistic
• If n

20
Test of µ, s Unknown, Population Normally
Distributed

• Test Statistic
• where
• is the sample statistic.
• µ0 is the value identified in the null
  hypothesis.
• s is unknown.
• n is the sample size
• degrees of freedom on t are n 1.

m
x

21
Test of µ, s Unknown, Population Not Normally
Distributed

• If n 30, Test Statistic
• If n

22
Test of p, Sample Sufficiently Large

• If both n p 5 and n(1 p) 5,
• Test Statistic
• where p sample proportion
• p0 is the value identified in the null
  hypothesis.
• n is the sample size.

23
Test of p, Sample Not Sufficiently Large

• If either n p proportion to the underlying binomial
  distribution.
• Note there is no t-test on a population
  proportion.

24
Observed Significance Levels

• A p-Value is
• the exact level of significance of the test
  statistic.
• the smallest value a can be and still allow us to
  reject the null hypothesis.
• the amount of area left in the tail beyond the
  test statistic for a one-tailed hypothesis test
  or
• twice the amount of area left in the tail beyond
  the test statistic for a two-tailed test.
• the probability of getting a test statistic from
  another sample that is at least as far from the
  hypothesized mean as this sample statistic is.

25
Observed Significance Levels

• A p-Value is
• the exact level of significance of the test
  statistic.
• the smallest value a can be and still allow us to
  reject the null hypothesis.
• the amount of area left in the tail beyond the
  test statistic for a one-tailed hypothesis test
  or
• twice the amount of area left in the tail beyond
  the test statistic for a two-tailed test.
• the probability of getting a test statistic from
  another sample that is at least as far from the
  hypothesized mean as this sample statistic is.

26
Several Samples

• Independent Samples
• Testing a companys claim that its peanut butter
  contains less fat than that produced by a
  competitor.

• Dependent Samples
• Testing the relative fuel efficiency of 10 trucks
  that run the same route twice, once with the
  current air filter installed and once with the
  new filter.

27
Test of (µ1 µ2), s1 s2, Populations Normal

• Test Statistic
• where degrees of freedom on t n1 n2 2

28
ExampleComparing Two populations
H0 pop1 pop2 H1 pop1 n.e. pop2
Hypothesis
Assumption
Test Statistic
The mean of population 1 is equal to the mean of
population 2
(1) Both distributions are normal (2) s1 s2
t distribution with df n1 n2-2
29
ExampleComparing Two populations
Rejection Region
Rejection Region
t distribution with df n1 n2-2
30
Test of (µ1 µ2), s1 n.e. s2, Populations
Normal, large n

• Test Statistic
• with s12 and s22 as estimates for s12 and s22

31
Test of Dependent Samples(µ1 µ2) µd

• Test Statistic
• where d (x1 x2)
• Sd/n, the average difference
• n the number of pairs of observations
• sd the standard deviation of d
• df n 1

32
Test of (p1 p2), where n1p15, n1(1p1)5,
n2p25, and n2 (1p2 )5

• Test Statistic
• where p1 observed proportion, sample 1
• p2 observed proportion, sample 2
• n1 sample size, sample 1
• n2 sample size , sample 2

33
Test of Equal Variances

• Pooled-variances t-test assumes the two
  population variances are equal.
• The F-test can be used to test that assumption.
• The F-distribution is the sampling distribution
  of s12/s22 that would result if two samples were
  repeatedly drawn from a single normally
  distributed population.

34
Test of s12 s22

• If s12 s22 , then s12/s22 1. So the
  hypotheses can be worded either way.
• Test Statistic whichever is larger
• The critical value of the F will be F(a/2, n1,
  n2)
• where a the specified level of significance
• n1 (n 1), where n is the size of the
  sample with the larger variance
• n2 (n 1), where n is the size of the
  sample with the smaller variance

35
Confidence Interval for (µ1 µ2)

• The (1 a) confidence interval for the
  difference in two means
• Equal variances, populations normal
• Unequal variances, large samples

36
Confidence Interval for (p1 p2)

• The (1 a) confidence interval for the
  difference in two proportions
• when sample sizes are sufficiently large.

37
Summary
Hypothesis
Assumption
Test Statistic
The mean of population 1 is equal to the mean of
population 2
(1) Both distributions are normal (2) s1 s2
The standard deviation of population 1 is equal
to the standard deviation of population 2
Both distributions are normal
The proportion of error in population 1 is equal
to the proportion of errors in population 2
n1p1 and n2p2 5 (approximation by normal
distribution)