Download Inference for Categorical Variable: More about Hypothesis Tests | STAT 0200 and more Study notes Statistics in PDF only on Docsity!
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture
Lecture 23
Inference for Categorical Variable:
More About Hypothesis Tests
Examples of Tests with 3 Forms of Alternative How Form of Alternative Affects Test When P -Value is “Small”: Statistical Significance Hypothesis Tests in Long-Run Relating Test Results to Confidence Interval (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 2
Looking Back: Review
4 Stages of Statistics Data Production (discussed in Lectures 1-4) Displaying and Summarizing (Lectures 5-12) Probability (discussed in Lectures 13-20) Statistical Inference 1 categorical: confidence intervals, hypothesis tests 1 quantitative categorical and quantitative 2 categorical 2 quantitative (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 3
Three Types of Inference Problem (Review)
In a sample of 446 students, 0.55 ate breakfast.
- What is our best guess for the proportion of all students who eat breakfast? Point Estimate
- What interval should contain the proportion of all students who eat breakfast? Confidence Interval
- Do more than half (50%) of all students eat breakfast? Hypothesis Test (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 4
Hypothesis Test About p (Review)
State null and alternative hypotheses and : Null is “status quo”, alternative “rocks the boat”.
- Consider sampling and study design.
- Summarize with , standardize to z , assuming that is true; consider if z is “large”.
- Find P -value=prob.of z this far above/below/away from 0; consider if it is “small”.
- Based on size of P -value, choose or.
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 5
Checking Sample Size: C.I. vs. Test
Confidence Interval: Require observed counts
in and out of category of interest to be at least
Hypothesis Test: Require expected counts in
and out of category of interest to be at least 10
(assume p = ).
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 6
Example: Checking Sample Size in Test
Background : 30/400=0.075 students picked #7 “at random” from 1 to 20. Want to test : p =0.05 vs.
. : p >0.05. Question: Is n large enough to justify finding P -value based on normal probabilities? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 8
Example: Checking Sample Size in Test
Background : 30/400=0.075 students picked #7 “at random” from 1 to 20. Want to test : p =0.05 vs.
. : p >0.05. Response: n = n (1- )=
Looking Back: For confidence interval, checked
30 and 370 both at least 10. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 9
Example: Test with “>” Alternative (Review)
Note : Step 1 requires 3 checks: Is sample unbiased? (Sample proportion has mean 0.05?) Is population ≥ 10 n? (Formula for s.d. correct?) Are np o and n (1- p o) both at least 10? (Find or estimate P -value based on normal probabilities?)
- Students are “typical” humans; bias is issue at hand.
- If p =0.05, sd of is and 3. P -value = is small: just over 0.
- Reject , conclude Ha: picks were biased for #7.
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 17
Example: Test with “Not Equal” Alternative
Background : 43% of Florida’s community college students are disadvantaged. Response: First write :_____ vs. : _____
- 356(0.43), 356(1-0.43) both≥10; pop.≥10(356)
- =____, z =_____ 3. P -value = ________________________
- Reject? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 19
Example: Test with “Not Equal” Alternative
Note: P -value is a two-tailed probability because alternative was “not equal” (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 20
90-95-98-99 Rule: “Outside” Probabilities
1.70 just over 1.645 P-value just under 2(.05) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 21
One-sided or Two-sided Alternative
Form of alternative hypothesis impacts
P -value
P -value is the deciding factor in test
Alternative should be based on what
researchers hope/fear/suspect is true
before “snooping” at the data
If < or > is not obvious, use two-sided
alternative (more conservative)
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 23
Example: How Form of Alternative Affects Test
Background : 43% of Florida’s community college students are disadvantaged. Question: Is % disadvantaged at Florida Keys (169/356=47.5%) unusually high? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 25
Example: How Form of Alternative Affects Test
Background : 43% of Florida’s community college students are disadvantaged. Response: Now write :______ vs. :_______
- Same checks of data production as before.
- Same =0.475, z =+1.70.
- Now P -value=_____________________________
- Reject? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 26
P -value for One- or Two-Sided Alternative
P -value for one-sided alternative is half
P -value for two-sided alternative.
P -value for two-sided alternative is twice
P -value for one-sided alternative.
For this reason, two-sided alternative is more
conservative (larger P -value, harder to
reject Ho).
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 27
Thinking About Data
Before getting caught up in details of test,
consider evidence at hand.
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 33
Example: Reviewing P-values and Conclusions
Background : Consider our prototypical examples: Are random number selections biased? P -value=0. Do fewer than half of commuters walk? P -value=0. Is % disadvantaged significantly different? P -value=0. Is % disadvantaged significantly higher? P -value=0. Question: What conclusions did we draw, based on those P -values? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 35
Example: Reviewing P-values and Conclusions
Background : Consider our prototypical examples: Are random number selections biased? P -value=0. Do fewer than half of commuters walk? P -value=0. Is % disadvantaged significantly different? P -value=0. Is % disadvantaged significantly higher? P -value=0. Response: (Consistent with 0.05 as cut-off ) P -value=0.011Reject? ___ P -value=0.299 Reject? ___ P -value=0.088Reject? ___ P -value=0.044Reject? ___ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 36
Example: Cut-Offs for “Small” P-Value
Background : Bookstore chain will open new
store in a city if there’s evidence that its
proportion of college grads is higher than
0.26, the national rate.
Question: Choose cut-off (0.10, 0.05, 0.01):
if no other info is provided if chain is enjoying considerable profits; owners are eager to pursue new ventures if chain is in financial difficulties, can’t afford losses if unsuccessful due to too few grads (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 38
Example: Cut-Offs for “Small” P-Value
Response: Choose cut-off (0.10, 0.05, 0.01):
if no other info is provided: use ___ if chain is enjoying considerable profits; owners are eager to pursue new ventures: use ___ if chain is in financial difficulties, can’t afford loss if unsuccessful due to too few grads: use ___
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 39
Definition
Statistically significant data: produce P -value small enough to reject. z plays a role: Reject if P -value small; if | z | large; if… Sample proportion far from Sample size n large Standard deviation small (if is close to 0 or 1) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 40
Role of Sample Size n
Large n : may reject even though
observed proportion isn’t very far from ,
from a practical standpoint.
Very small P -valuestrong evidence against
Ho but p not necessarily very far from p o.
Small n : may fail to reject even though
it is false.
Failing to reject false Ho is 2nd^ type of error
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 41
Definition
Type I Error: reject null hypothesis even
though it is true (false positive)
Probability is cut-off
Type II Error: fail to reject null
hypothesis even though it’s false
(false negative)
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 42
Hypothesis Test and Long-Run Behavior
Repeatedly carry out hypothesis tests of p =0.5,
based on 20 coinflips, using cut-off 5%.
In the long run, 5% of the tests will reject
. : p =0.5, even though it’s true.
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 48
Example: C.I. Results, Based on Test
Background : A hypothesis test did not reject
. : p =0.5 in favor of the alternative : p <0.5.
Question: Do we expect 0.5 to be contained
in a confidence interval for p?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L23. 50
Example: C.I. Results, Based on Test
Background : A hypothesis test did not reject
. : p =0.5 in favor of the alternative : p <0.5.
Response:
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L19. 51
Lecture Summary
(More Hypothesis Tests for Proportions)
Examples with 3 forms of alternative hypothesis Form of alternative hypothesis Effect on test results When data render formal test unnecessary P -value for 1-sided vs. 2-sided alternative Cut-off for “small” P -value Statistical significance; role of n , Type I or II Error Hypothesis tests in long-run Relating tests and confidence intervals
