Last Time:

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Last Time

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

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Four scenarios when making a decision based on a
sample
Great!
Type II Error
Great!
Type I Error
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Example
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100
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Example
?
?
2.5
-1.96
1.96
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Example
Alternatively, we could compute 95 Confidence
Interval for ?
100
100.5
100.892
100.108
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Example
How to compute (two-sided) p-value
2.5
-2.5
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Three equivalent methodsof hypothesis
testing(?significance level)
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Today

• Type II Error and Power
• Estimation Hypothesis Testing
• without the assumption that
• we know the population variance
• (or the population standard deviation)

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Hypothesis Testing as a Decision Problem
Great!
Type II Error
Power 1 P(Type II error) Our ability to
reject the null hypothesis when it is indeed
false
Great!
Type I Error
Depends on sample size and how much the null and
alternative hypotheses differ
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Type II Error

• Suppose we use the critical value method with a
  fixed significance level.

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Type II Error
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Type II Error
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Example Type II Error Power
Rejection Region for Null Hypothesis
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Example Type II Error Power
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Example Type II Error Power
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Another Example Type II Error Power
Rejection Region for Null Hypothesis
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Another Example Type II Error Power
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Another Example Type II Error Power
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Another Example Type II Error Power
Small Sample Size!
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Getting rid of one simplifying assumption
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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!
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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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WHAT IF ?X IS NOT KNOWN ?



Use the point estimate for ?X Find
sample variance S2 and use S as your best guess
for ?X.

?


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The student t distribution

• Looks like a Normal Distribution
• Depends on the sample size
• For sample size n the t-distribution
• has n-1 degrees of freedom
• t(n-1) ? N(0,1) as n approaches infinity
• Discovered by W.S.Gosset (Guinness)

http//www-stat.stanford.edu/naras/jsm/TDensity/T
Density.html
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IF ?X IS NOT KNOWN



P( Confidence Interval encloses ?X ) 1 - ?
Use the point estimate for ?X Find
sample variance S2 and use S as your best guess
for ?X. This affects the critical value
Have to use t-score t?/2, n-1
(see Table on last page of book)
n-1 of degrees of freedom (df) Note
t-score approaches z-score as n gets large
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IF ?X IS NOT KNOWN



Point estimate
Std. dev. of point estimate
critical value


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When the population variance is unknown
When the population variance is known
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Decision Tree for Confidence Intervals
n large? (CLT?)
X normal?
Population Variance known?
z-score
Yes
z-score
Yes
No
Yes
No
Cant do it
Yes
t-score
No
No
t-score
Yes
No
Cant do it
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ExampleConfidence Intervals Hypothesis
Testing when the Population Variance is unknown

• Suppose the birth weight of normal children has
  normal distribution with mean 115.2 oz.
• Suppose we collect a random sample of 20 babies
  from mothers who smoke.
• Suppose that
• the sample mean weight is 114 oz
• the sample standard deviation is 4.3.
• Is this significant evidence that babies of
  smokers differ in weight from normal ones?

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Lets compute a 95 CI
Sample does not provide significant evidence of
a difference in birth weight. (Two-sided test)
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ExampleConfidence Intervals Hypothesis
Testing when the Population Variance is unknown

• Suppose the birth weight of normal children has
  normal distribution with mean 115.2 oz.
• Suppose we collect a random sample of 20 babies
  from mothers who smoke.
• Suppose that
• the sample mean weight is 114 oz
• the sample standard deviation is 4.3.
• Is this significant evidence that babies of
  smokers are lighter than normal ones?

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Now we need a 1-sidedHypothesis Test
Whats the p-value of that standardized
statistic??
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Now we need a 1-sided Hypothesis Test
Whats the p-value of that standardized
statistic??
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Now we need a 1-sided Hypothesis Test
Whats the p-value of that standardized
statistic??
.080
.262
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Now we need a 1-sided Hypothesis Test
Whats the p-value of that standardized
statistic??
.080
.262