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Transcript
Chapter 2 –Basic Statistical Methods
• Describing sample data
–
–
–
–
–
Random samples
Sample mean, variance, standard deviation
Populations versus samples
Population mean, variance, standard deviation
Estimating parameters
• Simple comparative experiments
– The hypothesis testing framework
– The two-sample t-test
– Checking assumptions, validity
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Portland Cement Formulation (page 24)
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Graphical View of the Data
Dot Diagram, Fig. 2.1, pp. 24
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If you have a large sample, a
histogram may be useful
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Box Plots, Fig. 2.3, pp. 26
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The Hypothesis Testing Framework
• Statistical hypothesis testing is a useful
framework for many experimental
situations
• Origins of the methodology date from the
early 1900s
• We will use a procedure known as the twosample t-test
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The Hypothesis Testing Framework
• Sampling from a normal distribution
• Statistical hypotheses: H :   
0
1
2
H1 : 1   2
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Estimation of Parameters
1 n
y   yi estimates the population mean 
n i 1
n
1
2
2
S 
( yi  y ) estimates the variance 

n  1 i 1
2
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Summary Statistics (pg. 36)
Formulation 1
Formulation 2
“New recipe”
“Original recipe”
y1  16.76
y2  17.04
S  0.100
S22  0.061
S1  0.316
S2  0.248
n1  10
n2  10
2
1
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How the Two-Sample t-Test Works:
Use the sample means to draw inferences about the population means
y1  y2  16.76  17.04  0.28
Difference in sample means
Standard deviation of the difference in sample means
 
2
y
2
n
This suggests a statistic:
Z0 
y1  y2
 12
n1
Chapter 2

 22
n2
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How the Two-Sample t-Test Works:
Use S and S to estimate  and 
2
1
2
2
2
1
The previous ratio becomes
2
2
y1  y2
2
1
2
2
S
S

n1 n2
However, we have the case where     
2
1
2
2
2
Pool the individual sample variances:
(n1  1) S  (n2  1) S
S 
n1  n2  2
2
p
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2
1
2
2
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How the Two-Sample t-Test Works:
The test statistic is
y1  y2
t0 
1 1
Sp

n1 n2
• Values of t0 that are near zero are consistent with the null
hypothesis
• Values of t0 that are very different from zero are consistent
with the alternative hypothesis
• t0 is a “distance” measure-how far apart the averages are
expressed in standard deviation units
• Notice the interpretation of t0 as a signal-to-noise ratio
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The Two-Sample (Pooled) t-Test
(n1  1) S12  (n2  1) S22 9(0.100)  9(0.061)
S 

 0.081
n1  n2  2
10  10  2
2
p
S p  0.284
t0 
y1  y2
16.76  17.04

 2.20
1 1
1 1
Sp

0.284

n1 n2
10 10
The two sample means are a little over two standard deviations apart
Is this a "large" difference?
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William Sealy Gosset (1876, 1937)
Gosset's interest in barley cultivation led
him to speculate that design of
experiments should aim, not only at
improving the average yield, but also at
breeding varieties whose yield was
insensitive (robust) to variation in soil and
climate.
Developed the t-test (1908)
Gosset was a friend of both Karl Pearson
and R.A. Fisher, an achievement, for each
had a monumental ego and a loathing for
the other.
Gosset was a modest man who cut short
an admirer with the comment that “Fisher
would have discovered it all anyway.”
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The Two-Sample (Pooled) t-Test
• So far, we haven’t really
done any “statistics”
• We need an objective
basis for deciding how
large the test statistic t0
really is
• In 1908, W. S. Gosset
derived the reference
distribution for t0 …
called the t distribution
• Tables of the t
distribution – see
textbook appendix
Chapter 2
t0 = -2.20
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The Two-Sample (Pooled) t-Test
• A value of t0 between
–2.101 and 2.101 is
consistent with
equality of means
• It is possible for the
means to be equal and
t0 to exceed either
2.101 or –2.101, but it
would be a “rare
event” … leads to the
conclusion that the
means are different
• Could also use the
P-value approach
Chapter 2
t0 = -2.20
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The Two-Sample (Pooled) t-Test
t0 = -2.20
•
•
•
•
The P-value is the area (probability) in the tails of the t-distribution beyond -2.20 + the
probability beyond +2.20 (it’s a two-sided test)
The P-value is a measure of how unusual the value of the test statistic is given that the null
hypothesis is true
The P-value the risk of wrongly rejecting the null hypothesis of equal means (it measures
rareness of the event)
The P-value in our problem is P = 0.042
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Computer Two-Sample t-Test Results
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Checking Assumptions –
The Normal Probability Plot
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Importance of the t-Test
• Provides an objective framework for simple
comparative experiments
• Could be used to test all relevant hypotheses
in a two-level factorial design, because all
of these hypotheses involve the mean
response at one “side” of the cube versus
the mean response at the opposite “side” of
the cube
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Confidence Intervals (See pg. 44)
• Hypothesis testing gives an objective statement
concerning the difference in means, but it doesn’t
specify “how different” they are
• General form of a confidence interval
L    U where P( L    U )  1  
• The 100(1- α)% confidence interval on the
difference in two means:
y1  y2  t / 2,n1  n2 2 S p (1/ n1 )  (1/ n2 )  1  2 
y1  y2  t / 2,n1  n2 2 S p (1/ n1 )  (1/ n2 )
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Other Chapter Topics
• Hypothesis testing when the variances are
known
• One sample inference
• Hypothesis tests on variances
• Paired experiments
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