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Review for Test #2
Lecture 20
April 4, 2000
Prepare for Test #2
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Homework and test solutions are on the
course web page. Solutions and section
4.4 are in Adobe Acrobat (.pdf) format, a
downloadable copy of which can be found
online.
http://www.adobe.com/products/acrobat/readstep.htm
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Use section summaries
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Solve extra problems
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Sample test on web
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Make two pages of notes
Tables will be provided
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Office hours:
Prof: M 12 - 1 p.m. and Tu, W 3:30 - 4:30 p.m
TA: M 9 - 12, Tu 9 - 12 and Th 10-12
What to bring
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Calculator
Pencils
ID
2 pages of notes
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Tables will be provided
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Chapter 4
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Randomness
– Parameter
– Statistic
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Probability
– Sampling variability
– Independent trials
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Probability models
– Sample space
– Events
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Probability rules
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0  P(A)  1
P(S) = 1
P(A does not occur) = 1 - P(A)
If A and B are disjoint
P(A or B) = P(A) + P(B)
Assigning Probabilities
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Finite number of outcomes
– equally likely outcomes
– unequal probabilities
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Intervals of outcomes
– uniform distribution
– normal distribution
Sampling Distributions
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Law of large numbers
the observed mean outcome J must
approach the mean m of the population
as the number of observations increases
The sampling distribution of J describes
how J varies from sample to sample
Central limit theorem
for large n the sampling distribution of J is
approximately normal for any population
with finite standard deviation s
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N  ,s
n
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J Control Chart
To evaluate the control of a process with
given standards  and s, make an J control
chart as follows:
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Plot the means J of regular samples of
size n against time
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Draw a horizontal center line at 
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Draw horizontal control limits at  ±3s/n
Any J that does not fall between the control
limits is evidence that the process is “out of
control”
Statistical Inference
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Methods for drawing conclusions about a
population from sample data are called
statistical inference
Methods
1. Confidence Intervals - estimating a value
of a population parameter
2. Tests of significance - assess evidence
for a claim about a population
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Requirements:
1. The data must be an SRS from the
population
2. The data must be from either a normal
population or a large sample
3. We must know the standard deviation of
the population
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Confidence Intervals
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Confidence interval
estimate  margin of error
x z*
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s
n
Confidence level states the probability
that the method will give the correct
answer
Margin of error gets smaller as
– the confidence level decreases
– the population standard deviation s
decreases
– the sample size increases
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Sample size for a desired margin of error
 z *s 
n
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 m 
2
Stating hypotheses
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The null hypothesis H0 is a claim that we
will try to find evidence against
H0 is of the form H0:  =  0
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The alternative hypothesis Ha is the claim
about the population that we are trying to
find evidence for
The alternative hypothesis can be onesided if we are interested in deviations
from the null hypothesis in one direction
Ha is of the form H0:  >  0
or H0:  <  0
The alternative hypothesis can be twosided if we are interested in any deviation
from the null hypothesis
Ha is of the form Ha:    0
Test Statistic
A test statistic measures the compatibility
between the null hypothesis and the data.
Because normal calculations require
standardized variables, we use as our
test statistic the standardized sample
mean
x  0
z
s n
This random variable has the standard
normal distribution N(0, 1) when the null
hypothesis is true.
P-value
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The P-value is the probability of getting
an outcome as extreme or more extreme
than the actually observed outcome
The probability is computed assuming
that H0 is true
The smaller the P-value, the stronger the
evidence against H0 provided by the data
If the P-value is as small or smaller than
the significance level a, we say that the
data are statistically significant at level a.
Test of significance
1. State the null hypothesis H0 and the
alternative hypotheses Ha.
The test is designed to asses the strength
of evidence against H0; Ha is the
statement we will accept if we reject H0
2*. Select a significance level.
3. Calculate the test statistic.
4. Find the P-value for the observed data.
5. State a conclusion.
If P-value  a reject Ho in favor of Ha
If P-value > a accept Ho
Cautions
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There is no sharp border between
significant and insignificant results
Statistical significance can be obtained by
increasing the sample size
Practical significance must be considered
Data must come from an SRS and be
either normally distributed or from a large
sample
Beware of multiple analyses
Errors
We can make two types of errors:
Type I - we reject H0 (accept Ha) when in
fact H0 is true
This happens if we reject the lot (based
on the sample) and the lot really does
meet standards
Type II - we accept H0 (reject Ha) when in
fact Ha is true
This happens if we accept the lot (based
on the sample) and the lot really does not
meet standards
a = P(Type I error)
b = P(Type II error)
Power = 1 - b