Transcript of Statistics: Unlocking the Power of Data Lock 5 Inference for Proportions STAT 250 Dr. Kari Lock...
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- Statistics: Unlocking the Power of Data Lock 5 Inference for
Proportions STAT 250 Dr. Kari Lock Morgan Chapter 6.1, 6.2, 6.3,
6.7, 6.8, 6.9 Formulas for standard errors Normal based
inference
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- Statistics: Unlocking the Power of Data Lock 5 Confidence
Interval Formula From original data From bootstrap distribution
From N(0,1) IF SAMPLE SIZES ARE LARGE
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- Statistics: Unlocking the Power of Data Lock 5 Formula for
p-values From randomization distribution From H 0 From original
data Compare z to N(0,1) for p-value IF SAMPLE SIZES ARE LARGE
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- Statistics: Unlocking the Power of Data Lock 5 Standard Error
Wouldnt it be nice if we could compute the standard error without
doing thousands of simulations? We can!!!
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- Statistics: Unlocking the Power of Data Lock 5
ParameterDistributionStandard Error Proportion Normal Difference in
Proportions Normal Meant, df = n 1 Difference in Meanst, df = min(n
1, n 2 ) 1 Correlationt, df = n 2 Standard Error Formulas
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- Statistics: Unlocking the Power of Data Lock 5 SE Formula
Observations n is always in the denominator (larger sample size
gives smaller standard error) Standard error related to square root
of 1/n Standard error formulas use population parameters (uh oh!)
For intervals, plug in the sample statistic(s) as your best guess
at the parameter(s) For testing, plug in the null value for the
parameter(s), because you want the distribution assuming H 0
true
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- Statistics: Unlocking the Power of Data Lock 5 Hormone
Replacement Therapy Until 2002, hormone replacement therapy (HRT),
estrogen and/or progesterone, was commonly prescribed to
post-menopausal women. This changed in 2002, when the results of a
large clinical trial were published 8506 women were randomized to
take HRT, 8102 were randomized to placebo. 166 HRT and 124 placebo
women developed invasive breast cancer Does hormone replacement
therapy cause increased risk of breast cancer?
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- Statistics: Unlocking the Power of Data Lock 5 Interval for
Proportion First: What are the chances a woman not taking HRT
develops invasive breast cancer? Give a 90% confidence interval.
What is the sample statistic? What is z* for a 90% interval? What
is the standard error?
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- Statistics: Unlocking the Power of Data Lock 5 Sample Statistic
8506 women were randomized to take HRT, 8102 were randomized to
placebo. 166 HRT and 124 placebo women developed invasive breast
cancer
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- Statistics: Unlocking the Power of Data Lock 5 z*
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- Statistics: Unlocking the Power of Data Lock 5 Standard
Error
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- Statistics: Unlocking the Power of Data Lock 5 Standard
Error
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- Statistics: Unlocking the Power of Data Lock 5 Interval for
Proportion
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- Statistics: Unlocking the Power of Data Lock 5 Bootstrap
Interval
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- Statistics: Unlocking the Power of Data Lock 5 Interpretation
We are 90% confident that the true proportion of post-menopausal
women not taking HRT who develop invasive breast cancer is between
0.013 and 0.018 Do you believe this sentence is accurate? a) Yes b)
No
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- Statistics: Unlocking the Power of Data Lock 5 Your Turn! 8506
women were randomized to take HRT, 8102 were randomized to placebo.
166 HRT and 124 placebo women developed invasive breast cancer Give
a 95% confidence interval for the proportion of women taking HRT
who develop invasive breast cancer: a) (0.013, 0.027) b) (0.015,
0.025) c) (0.017, 0.023) d) (0.019, 0.021)
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- Statistics: Unlocking the Power of Data Lock 5 Testing When
testing, use the null value for p, rather than the sample statistic
So, if testing whether the proportion of people not taking HRT who
develop invasive breast cancer is less than 0.1, the standard error
would be
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- Statistics: Unlocking the Power of Data Lock 5 Hormone
Replacement Therapy 8506 women were randomized to take HRT, 8102
were randomized to placebo. 166 HRT and 124 placebo women developed
invasive breast cancer Does hormone replacement therapy cause
increased risk of breast cancer?
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- Statistics: Unlocking the Power of Data Lock 5 Hypothesis Test
State hypotheses. Calculate z-statistic. Compare to standard normal
distribution to find p-value. Make a conclusion.
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- Statistics: Unlocking the Power of Data Lock 5 Hypotheses Does
hormone replacement therapy cause increased risk of invasive breast
cancer? p 1 = proportion of women taking HRT who get invasive
breast cancer p 2 = proportion of women not taking HRT who get
invasive breast cancer a) H 0 : p 1 = p 2, H a : p 1 p 2 b) H 0 : p
1 = p 2, H a : p 1 > p 2 c) H 0 : p 1 = p 2, H a : p 1 < p
2
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- Statistics: Unlocking the Power of Data Lock 5 Calculate
z-statistic null value = 0 What to use for p 1 and p 2 ??? Have to
assume null is true
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- Statistics: Unlocking the Power of Data Lock 5 Null Values
Testing a difference in proportions: H 0 : p 1 = p 2 Use the
overall sample proportion from both groups (called the pooled
proportion) as an estimate for both p 1 and p 2 Note that this is
in between the sample proportions for each group.
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- Statistics: Unlocking the Power of Data Lock 5 Standard
Error
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- Statistics: Unlocking the Power of Data Lock 5 Standard
Error
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- Statistics: Unlocking the Power of Data Lock 5 z-statistic
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- Statistics: Unlocking the Power of Data Lock 5 p-value
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- Statistics: Unlocking the Power of Data Lock 5 p-value
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- Statistics: Unlocking the Power of Data Lock 5 Conclusion Does
this provide evidence that hormone replacement therapy increases
risk of invasive breast cancer? a) Yes b) No
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- Statistics: Unlocking the Power of Data Lock 5 Your Turn! Same
trial, different variable of interest. 8506 women were randomized
to take HRT, 8102 were randomized to placebo. 502 HRT and 458
placebo women developed any kind of cancer. Does hormone
replacement therapy cause increased risk of cancer in general? a)
Yes b) No
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- Statistics: Unlocking the Power of Data Lock 5 Margin of Error
For a single proportion, what is the margin of error? a) b) c)
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- Statistics: Unlocking the Power of Data Lock 5 Margin of Error
You can choose your sample size in advance, depending on your
desired margin of error! Given this formula for margin of error,
solve for n.
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- Statistics: Unlocking the Power of Data Lock 5 Margin of
Error
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- Statistics: Unlocking the Power of Data Lock 5 Margin of Error
Suppose we want to estimate a proportion with a margin of error of
0.03 with 95% confidence. How large a sample size do we need?
(a)About 100 (b)About 500 (c)About 1000 (d) About 5000
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- Statistics: Unlocking the Power of Data Lock 5 To Do Read
Sections 6.1, 6.2, 6.3, 6.7, 6.8, 6.9 Do HW 6a (due Friday,
4/10)