Statistical Inference

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Statistical Inference

• Trey Spencer
• Division of Biometry

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Statistical Inference
Two approaches are commonly used in making
statements about population parameters Estimatio
n, and Hypothesis testing.
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Estimation

• The goal of estimation is to describe values of
• population parameters using information gained
  from samples.
• A point estimate is a value estimated from a
  sample, for example, a sample mean.
• An interval estimate is a range of values that is
  reasonably certain to contain the value of the
  parameter of interest, for example, a confidence
  interval for population mean.

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Estimation

• Characteristics of a good point estimator
• The sampling distribution of the point estimator
  should be centered over the true value of the
  parameter to be estimated.
• The spread (as measured by the variance) of the
  sampling distribution should be as small as
  possible.

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Estimation
Biased estimate of center, moderate variance
Unbiased estimate of center, but large variance
Unbiased estimate and small variance
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Hypothesis Tests

• Hypothesis Testing makes inferences about
  (population) parameters by supposing they have
  certain values, and then testing whether the
  observed data is consistent with the hypothesis.

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Hypothesis TestsDefinitions

• The null hypothesis, HO, usually states that the
  observed results are coincidental, i.e. there is
  no real effect.
• The alternative hypothesis, HA, states that the
  observations are the results of a real effect,
  plus random variation.

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Hypothesis TestsDefinitions

• The Test Statistic is a statistic used in
  carrying out the test of a hypothesis.
• The Rejection Region consist of the set of values
  of a statistic for which the null hypothesis is
  rejected.
• The Critical Value is the value, or values
  separating the rejection region and the
  acceptance region.

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Hypothesis TestsDefinitions
A Right-Tail Test is a test where the rejection
region is in the right tail of the distribution
of the test statistic. A Left-Tail Test is a
test where the rejection region is in the left
tail of the distribution of the test
statistic. A Two-Tail Test is a test where the
rejection region is in the right and the left
tail of the distribution of the test statistic..
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Hypothesis TestsDefinitions

• 1). Right-Tail Test
• HO µ ? µ0
• HA µ µ0
• 2). Left-Tail Test
• HO µ ? µ0
• HA µ
• 3). Two-Tail Test
• HO µ µ0
• HA µ ? µ0

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Hypothesis Tests
Critical Values
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Hypothesis Tests
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Hypothesis TestsDefinitions

• Type I Error The error of rejecting H0 when H0
  is true. The probability of this error is called
  Level of Significance ?.
• Type II Error The error of accepting H0 when HA
  is true. The probability of this error is
  denoted as ?.
• Power The power of a test is the probability of
  rejecting the null hypothesis when it is false.
  It is denoted as 1 ?.

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Hypothesis Tests

• P-value The P-value is the probability under the
  null hypothesis of observing a value as unlikely
  or more unlikely, than was observed. It is
  calculated assuming that H0 is true and taking HA
  into account in determining what is meant by
  unlikely.

Guidelines for Interpreting P-values
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Hypothesis Tests
Steps involved in performing hypothesis
tests 1. Specify the hypotheses. 2.
Determine the appropriate test-statistic. 3.
Determine the critical value(s) and define the
rejection region. 4. Draw conclusions.
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One-Sample Inference
A (1- a )100 Large-Sample Confidence Interval
for a Population Mean m
When ? is known this equation is appropriate
regardless of sample size

• where za / 2 is the value corresponding to an
  area a / 2 in the upper tail of a standard normal
  distribution.
• If s is unknown, it can be approximated by the
  sample standard deviation s when the sample size
  is large (n 30) and the approximate confidence
  interval is

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One-Sample Inference
Testing Hypotheses about ?. H0??0 If ? is
known then use the test statistic
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One-Sample Inference
Testing Hypotheses about ?. H0??0 If ? is
unknown and n is large (30) then use the test
statistic
(Replace ? with s)
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One-Sample Inference
A (1- a )100 Small-Sample Confidence Interval
for a Population Mean m when s is unknown.
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One-Sample Inference
Small sample size and ? unknown under normality
assumption
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One-Sample Inference
Critical Values for the t Distribution
This table contains critical values t?,? for the
t distribution defined by P(T?t?,?) ?.
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One-Sample InferenceMatched Pairs

• With a paired sample (X1,Y1), , (Xn,Yn),
• calculate the difference DiXiYi, the mean D and
  the standard deviation SD. Based on the sample
  size, follow either the large or small one-sample
  procedures

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One-Sample InferenceMatched Pairs Example
Below is data acquired by Mazze el al. (1971)
that deals with the pre-operative and
post-operative creatinine clearance (ml/min) of
six patients anesthetized by halothane. Is there
evidence to suggest a change in creatinine
clearance has occurred?
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One-Sample InferenceMatched Pairs Example
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Hypothesis Tests
Steps involved in performing hypothesis
tests 1. Specify the hypotheses. 2.
Determine the appropriate test-statistic. 3.
Determine the critical value(s) and define the
rejection region. 4. Draw conclusions.
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One-Sample InferenceMatched Pairs Example
Step 1. State the hypotheses to be tested HO
µD µ0 0 HA µD ? µ0 0 Step 2. Determine
the appropriate test statistic n 6 xD
24.3 ml/min sD 42.2 µ00 t 1.42
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One-Sample InferenceMatched Pairs Example
Step 3. Determine critical values and Rejection
Region
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One-Sample InferenceMatched Pairs Example
95 Confidence Interval Around the Difference
The confidence interval indicates that the
difference between post- and pre-surgery
creatinine clearance could be as small as -19.97
ml/min or as much as 68.59 ml/min with 95
confidence.
(-19.97, 68.59)
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Two-Sample Inference
Large Sample Estimate of ( m 1 - m 2) Large
Sample Test of ( m 1 - m 2) Small Sample Estimate
of ( m 1 - m 2) Small Sample Test of ( m 1 - m 2)
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A Large Sample (1-a )100 Confidence Interval
for (m 1 - m 2 )
If ?1 and ?2 are unknown, they can be
approximated by the sample standard deviations s1
and s2 and the approximate confidence interval is
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Large-Sample Test for ( m 1 - m 2)

• 1. Null hypothesis H 0 ( m 1 - m 2) D 0
• Alternative hypothesis
• One-Tailed Test Two-Tailed Test
• H a ( m 1 - m 2) D 0 H a ( m 1 - m 2) ¹
  D 0
• H a ( m 1 - m 2)
• 2. Test statistic

If ?1 and ?2 are unknown, then use s1 and s2
3. Rejection region Reject H 0
when One-Tailed Test Two-Tailed Test z
za z za/2 or z - za/2 z
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A Small Sample (1-a )100 Confidence Interval
for (m 1 - m 2 )
where,
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Small-Sample Test for ( m 1 - m 2)

• 1. Null hypothesis H 0 ( m 1 - m 2) D 0
• Alternative hypothesis
• One-Tailed Test Two-Tailed Test
• H a ( m 1 - m 2) D 0 H a ( m 1 - m 2) ¹
  D 0
• H a ( m 1 - m 2)
• 2. Test statistic

3. Rejection region Reject H 0
when One-Tailed Test Two-Tailed Test t
ta t ta/2 or t - ta/2 t
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Another Example
Columns Variable
Abbreviation 2-4
Identification Code
ID 10 Low Birth Weight (0
Birth Weight 2500g, LOW
1 Birth Weight Age of the Mother in Years
AGE 23-25 Weight in Pounds
at the Last Menstrual Period LWT
32 Race (1 White, 2 Black, 3
Other) RACE 40
Smoking Status During Pregnancy (1 Yes, 0 No)
SMOKE 48 History of Premature
Labor (0 None, 1 One, etc.) PTL 55
History of Hypertension (1 Yes, 0 No)
HT 61 Presence of
Uterine Irritability (1 Yes, 0 No) UI
67 Number of Physician Visits During
the First Trimester FTV (0
None, 1 One, 2 Two, etc.) 73-76
Birth Weight in Grams
BWT
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Another Example
x1 135.2 s1 29.85
x2 121.6 s2 24.99
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Another Example
Step 1. State the hypotheses to be tested HO
µ1 - µ2 0 HA µ1 - µ2 ? 0 Step 2.
Determine the appropriate test statistic n1
26 n2 17 x1 135.2 lbs. x2 121.6 lbs.
s1 29.9 lbs. s2 25.0 lbs.
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Another Example
Step 3. Determine critical values and Rejection
Region
P-value 0.13
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Another Example
95 CI of the Difference in Group Means
-2.02
95 CI (-4.16, 31.18)