Due in Lab Sections Oct 31 / Nov 1

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Graded Homework Assignment

• 6.13 6.28 6.54 6.56 6.68
• Due in Lab Sections Oct 31 / Nov 1

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Grading
Regrading of any exam part means Regrading of
full exam. This may improve or worsen your
score. Any grievances with Homework Scores or
Quiz Scores up to today must be communicated to
Ying by the end of next week.
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Last Class and Today

• Estimation
• Point Estimates and
• Confidence Intervals

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Try to hit center of target (population parameter)
Shoot a bullet (sample statistic)
Center of Target
Bullet
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Sampling
We have already learned how precise the bullet
tends to be This was captured by the sampling
distribution of our sample statistic
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Confidence Intervals
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Confidence Intervals
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Example What is the average age of entering
students in US business schools? Population all
entering students in US Business Schools Want ?X
Collect a random sample. Suppose that
n330, 28.5 ? How confident can I be
that ?X is between 28 and 29? ? Which B would
provide 95 confidence that ?X is between
28.5-B and 28.5B? ? How large should the sample
be if I want to be 90 confident that ?X is
within 1 year from the value of the sample mean?
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Suppose we know that ?X 3 What is the
probability that is not more than .5 from
?X ? Recall So,
N(?X , .1651)
N(0,1)
-3.03
-3.03
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Suppose we know that ?X 3 What is the
probability that is not more than .5 from
?X ? Recall So,
N(?X , .1651)
99.76 confidence Interval for ?X
28
29
28.5
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New slide
Suppose we know that ?X 3 Suppose that ?X
28. What is the probability that is more
than 28.5?
Given a population mean of 28, the probability of
a sample mean further to the right than 28.5
is .0012
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99.76 confidence Interval for ?X
28.5
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New slide
Suppose we know that ?X 3 Suppose that ?X
29. What is the probability that is less
than 28.5?
Given a population mean of 29, the probability of
a sample mean further to the left than 28.5
is .0012
28
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99.76 confidence Interval for ?X
28.5
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New slide
Suppose we know that ?X 3 What is the
probability that is not more than .5 from
?X ?
28
29
The probability that we observe a sample mean
that deviates from the population mean by more
then ½ year is .0024 1-.9976 (2)(.0012)
28.5
99.76 confidence Interval for ?X
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Suppose we know that ?X 3 How large should
interval around be in order to provide 95
confidence that such interval, ( -B, B),
encloses ?X ? Recall N(?X ,
.1651) Want B such that
Standardize
N(0,1)
Extra work the table 1.96 B/.1651. So, B
.323596
1.96
-1.96
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Suppose we know that ?X 3 How large should
interval around be in order to provide 95
confidence that such interval, ( -B, B),
encloses ?X ? Recall N(?X ,
.1651) Want B such that
95 confidence Interval for ?X
28.28
28.82
28.5
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Suppose we know that ?X 3 How large should
interval around be in order to provide 95
confidence that such interval, ( -B, B),
encloses ?X ? Recall N(?X ,
.1651) Want B such that
New slide
95 confidence Interval for ?X
The probability that we observe as sample mean
that deviates from the population mean by more
then .32 years is .05 1-.95 (2)(.025)
28.28
28.82
28.5
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Suppose we know that ?X 3 How large should the
sample be, if we want to be 90 confident that (
-1, 1), encloses ?X ? Recall Want
smallest n such that
Standardize
N(0,1)
Extra work the table 1.645 So, ngt 24.35,
i.e., n25.
1.645
-1.645
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Today

• More on
• Confidence Intervals
• Start Hypothesis Testing

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Suppose we know that ?X 3 How large should the
sample be, if we want to be 90 confident that (
-1, 1), encloses ?X ? Recall Want
smallest n such that
90 confidence Interval for ?X
27.5
29.5
28.5
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New slide
Suppose we know that ?X 3 How large should the
sample be, if we want to be 90 confident that (
-1, 1), encloses ?X ? We concluded n25 to
get
90 confidence Interval for ?X
The probability that we observe as sample mean
that deviates from the population mean by more
then .5 years is .10 1-.90 (2)(.05)
27.5
29.5
28.5
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New slide
Note that we answered all three questions
without using the sampled value of . ?
How confident can I be that ?X is between -.5
and .5? ? Which B would provide 95
confidence that ?X is between -B and
B? ? How large should the sample be if I
want to be 90 confident that ?X is within
1 from the value of the sample mean?
B .323596
n25
You could address all three questions in a grant
or dissertation proposal before collecting the
data!
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CONFIDENCE INTERVAL 100(1 - ?) confidence
interval for a population parameter
Point estimate
Std. dev. of point estimate
critical value


P( C. I. encloses true population parameter )
1 - ? Note ? P(Confidence Interval misses true
population parameter ) Proportion of times such
a CI misses the population parameter
Parameters confidence level 1- ? critical
value. std. dev. of point estimate
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CONFIDENCE INTERVAL FOR ?X WHEN ?X KNOWN 100(1 -
?) confidence interval



P( Confidence Interval encloses ?X ) 1 - ?
Note ? P(Confidence Interval
misses ?X )
Note the relationship between ? probability
of missing ?X n sample size
half-width of the
confidence interval margin of
error Fix any two and the third is determined!
?X
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CONFIDENCE INTERVAL FOR ?X WHEN ?X KNOWN 100(1 -
?) confidence interval



Margin of Error

• Confidence Intervals are wide if
• population variance of X is large
• sample size is small
• Wide CI means that we are not very precise in
  estimating
• the population parameter (here the population
  mean)

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CONFIDENCE INTERVAL FOR ?X WHEN ?X KNOWN
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CONFIDENCE INTERVAL FOR ?X WHEN ?X KNOWN
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CONFIDENCE INTERVAL FOR ?X WHEN ?X KNOWN
N(0,1)
1.645
-1.645
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CONFIDENCE INTERVAL FOR ?X WHEN ?X KNOWN
N(0,1)
1.960
-1.960
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CONFIDENCE INTERVAL FOR ?X WHEN ?X KNOWN
N(0,1)
2.576
-2.576
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?X?
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?X?
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?X?
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?X?
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?X?
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Confidence interval for ?X
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Caution!
Confidence interval
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Lets get started

• Hypothesis Testing
• After Sampling and Estimation,
• yet another way to look at essentially the same
  information

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Example

• Lets consider once more the age of entering
  students in U.S. B-schools.
• Suppose (for simplicity) that ?X 3.
• An old government publication says the average
  age is ?X 28. You think that it is higher and
  you want to check your intuition statistically.
• You collect a random sample of size n330.
• It turns out in the sample the average age is
  28.5.

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Example

• Claim of the Government Publication ?X 28
• Null Hypothesis H0 ?X 28
• (Status Quo, Common Wisdom)
• Alternative Hypothesis Ha ?X gt 28
• (Your claim)
• Rules of the game
• Unless you can provide a lot of evidence for your
  claim, the status quo will prevail.
• The null hypothesis will be retained until enough
  evidence has been accumulated against it.
• In particular, all calculations will assume
  that ?X 28. (!!!)

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Null Hypothesis (Status Quo)

• Average entering age is 28 (until proven
  different)
• New product no different from old one (until
  proven better)
• Experimental group is no different from control
• group (until proven different)
• The accused is innocent (until proven
  guilty)

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Innocent
Guilty

• ? Ha is the hypothesis you are gathering evidence
  in support of.
• ? H0 is the fallback option the hypothesis you
  would like to reject.
• ? Reject H0 only when there is lots of evidence
  against it.
• A technicality always include in H0
• H0 (with sign) is assumed in all mathematical
  calculations!!!

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In our Example
1,000,000 Question How much evidence do we have
against Ho? Assuming Ho, how unusual is the
sample mean 28.5?
Recall Suppose we know that ?X 3. Suppose
that ?X 28. What is the probability that
is more than 28.5? (n330)
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Suppose we know that ?X 3. Ho ?X 28,
Ha ?X gt 28 What is the probability that
is more than 28.5? (n330)
The p-value of this hypothesis test is
.0012 Given that the Null Hypothesis holds, it
is extremely unlikely to observe a sample mean
equal or larger the one we obtained. ? REJECT Ho
Given a population mean of 28, the probability of
a sample mean further to the right than 28.5
is .0012
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Hypothesis Testing
p-value
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Hypothesis Testing
p-value
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Hypothesis Testing
p-value