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Sample proportion
- p-hat = y/n
- y = success
- n = number of trials
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Wilson-adjusted proportion
- p-tilda = (y+2)/(n+4)
- to keep p-tilda from 0 or 1
-
SE for p-hat
√(p-hat (1-p-hat)/n)
-
CI for p-hat (95%)
- p-hat +/- z.025√(p-hat(1-p-hat)/n) =
- p-hat +/- 1.96√(p-hat(1-p-hat)/n)
- set upper limit to 1 and lower limit to 0 if surpass
- unstable, sometimes over coverage, sometimes less (not always 95%)
-
Wilson adjusted 95% CI
- p-tilda +/- 1.96√(p-tilda(1-p-tilda)/(n+4))
- set upper limit to 1 and lower limit to 0
- Gives better coverage (closer to 95%)
-
One sided confidence interval
- (-∞,p-tilda + 1.65 * SEp-tilda)
- (p-tilda - 1.65 * SEp-tilda, ∞)
- Still between 0 or 1
-
Wilson SE
√(p-tilda(1 - p-tilda)/ (n + 4))
-
Χ2 for more than 2
- (Observed - Expected)2 / Expected + all values
- Use df
- All expected have to be greater than 5
- Can just say if they are different than expected
- Observed-expect2 will always be the same so
-
X2 for 2
- directional
- could use binomial
- H0: p = .75
- HA: p ≠ .75
- check with table
- For one sided, ts has to be >/< than 2*alpha ts AND on the right side of expected
-
Test for independence w/ contingency tables
- p1 = (A|B)
- p2 = (A|C)
- H0: p1 = p2
- HA: p1 = p2 (p1 >< p2)
-
Expected values in 2x2 tables
- row total * column total / Grand total
- Make sure each is at least 5
-
df in 2x2 tables
(# rows - 1) * (# columns - 1)
-
Directional test with X2 and 2x2
- X2 > Xtablefor 2alpha
- and
- Alternate hypothesis was satisfied
- Non-directional don't double
-
Interpretation for X2
- association not causal
- maybe causal in controlled study
- if one H0 is rejected, differently defined p will be also be rejected from same table
-
What is significance level
the likely-hood of making a type I error
-
CI for p-tilda
- p-tilda1 - p-tilda2 +/- ZApha/2 * SEp1-p2
- If it contains 0 no differences
-
SE for p-tilda
- To keep it away from 0
-
Assumptions for ANOVA
- each population is normally distributed
- samples are independent
- samples are random
-
For several categories why not pair-wise t-test?
- Because chance of committing type I error is large for whole test
- alpha for each pair
- 1- (1-alpha)# colums
- Problem of multiple comparisons
- if lower alpha get higher type II
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