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(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture
Lecture 25
Inference for Quantitative Variable:
Hypothesis Tests
z Test about Population Mean: 4 Steps Examples: 1-sided or 2-sided Alternative Relating Test and Confidence Interval Factors in Rejecting Null Hypothesis Inference Based on t vs. z (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 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 (discussed in Lectures 21-23) 1 quantitative: confidence intervals, hypothesis tests categorical and quantitative 2 categorical 2 quantitative (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 3
Three Types of Inference Problem
Mean yearly earnings for sample of 446 students at a particular university was $3776.
- What is our best guess for the mean earnings of all students at that university? (Point Estimate)
- What interval should contain mean earnings for all the students? (Confidence Interval)
- Is this convincing evidence that mean earnings for all the students is less than $5000? (Hypothesis Test) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 4
Behavior of Sample Mean (Review)
For random sample of size n from population
with mean and standard deviation ,
sample mean has
mean
standard deviation
shape approximately normal for large
enough n
If is known, standardized follows
z (standard normal) distribution
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 5
Hypothesis Test About (with z )
Problem Statement
- Consider sampling and study design.
- Summarize with , standardize to assuming is true; is z “large”?
- Find P -value (prob. of Z this far above/below/away from 0); is it “small”?
- Based on size of P -value, choose or. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 6
Hypothesis Test About with z (Details)
- Consider sampling and study design.
- Summarize with , standardize to assuming is true; is z “large”?
- Find prob. of z this far above/below/away from 0 ( P -value); consider if it is “small”.
- Based on size of P -value, choose or. If sample is biased, mean of is not. If pop<10 n , s.d. of is not. If n is too small, distribution of is not normal, won’t standardize to z : graph data, see guidelines (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 7
Hypothesis Test About with z (Details)
- Consider sampling and study design.
- Summarize with , standardize to assuming is true; is z “large”?
- Find prob. of z this far above/below/away from 0 ( P -value); consider if it is “small”.
- Based on size of P -value, choose or. Assess P -value based on form of alternative hypothesis (greater, less, or not equal) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 8
Hypothesis Test About with z (Details)
Alternative “>”: P -value is right-tailed probability
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 15 Example: Test with One-Sided Alternative Background: Earnings of 446 surveyed university students had mean $3776. Assume pop. s.d. $6500. Response:
- Students representative in terms of earnings?…
- Output shows sample mean 3.776 (thou $), z =-3.98. 3. P -value = ___________________ Small? _____
- Reject? ____ Conclude _________________ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 17 Example: Test with One-Sided Alternative Note: P -value is a left-tailed probability because alternative was “less than”. Response:
- Students representative in terms of earnings?…
- Output shows sample mean 3.776 (thou $), z =-3.98. 3. P -value =
- Reject? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 18 Example: Notation Background : Want to test if mean of all male shoe sizes could be 11.0, based on a sample mean 11. from 9 male students. Assume pop. s.d. 1.5. Question: How do we denote the numbers given? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 20 Example: Notation Background : Want to test if mean of all male shoe sizes could be 11.0, based on a sample mean 11. from 9 male students. Assume pop. s.d. 1.5. Response: 11.0 is proposed value of population mean ____ 11.222 is sample mean ____ 9 is sample size ____ 1.5 is population standard deviation ____
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 21 Example: Intuition Before Formal Test Background : Want to test if mean of all male shoe sizes could be 11.0, based on a sample mean 11. from 9 male students. Assume pop. s.d. 1.5. Question: What conclusion do we anticipate, by “eye-balling” the data? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 23 Example: Intuition Before Formal Test Background : Want to test if mean of all male shoe sizes could be 11.0, based on a sample mean 11. from 9 male students. Assume pop. s.d. $6500. Response: Sample mean (11.222) seems close to proposed =11.0? ___ Sample size (9) small_____________________________ S.d. (1.5) not very small___________________________ Anticipate standardized sample mean z large? _____ P -value small? _________ conclude population mean ________ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 24 Example: Test with Two-Sided Alternative Background: Want to test if mean of all male shoe sizes could be 11.0, based on a sample mean 11. from 9 male students. Assume pop. s.d. 1.5. Question: What do we conclude from the output? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 26 Example: Test with Two-Sided Alternative Background: Want to test if mean of all male shoe sizes could be 11.0, based on a sample mean 11. from 9 male students. Assume pop. s.d. 1.5. Response: z = 0.44. Large? _____ P -value (two-tailed) = 0.657. Small? _____ Conclude pop mean may be 11.0? _____
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 31
Example: Test Results and Confidence Interval
Background: Tested if mean of all male shoe sizes could be 11.0, based on a sample mean 11.222 from 9 male students. Assumed pop. s.d. 1.5. P -value was 0.657; did not reject Ho. Question: Would we expect 11.0 to be contained in a confidence interval for? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 33
Example: Test Results and Confidence Interval
Background: Tested if mean of all male shoe sizes could be 11.0, based on a sample mean 11.222 from 9 male students. Assumed pop. s.d. $6500. P -value was 0.657; did not reject. Response: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 34
Example: Test Results and Confidence Interval
Background: Tested if mean earnings of all students at a university could be $5000, based on a sample mean $3776 for n =446. Assumed pop. s.d. $6500. P -value was 0.000; rejected Ho. Question: Would 5000 be contained in the confidence interval for? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 36
Example: Test Results and Confidence Interval
Background: Tested if mean earnings of all students at a university could be $5000, based on a sample mean $3776 for n =446. Assumed pop. s.d. $6500. P -value was 0.000; rejected Ho. Response:
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 37 Factors That Lead to Rejecting Ho
Statistically significant data produce P -value
small enough to reject. z plays a role:
Reject if P -value small; if | z | large; if…
Sample mean far from
Sample size n large
Standard deviation small
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 38 Role of Sample Size n
Large n : may reject even if sample
mean is not far from proposed population
mean, from a practical standpoint.
Very small P -valuestrong evidence against
Ho but not necessarily very far from.
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 L25. 39 Definition (Review)
Type I Error: reject null hypothesis even
though it is true (false positive)
Type II Error: fail to reject null
hypothesis even though it’s false
(false negative)
Test conclusions determine possible error:
Reject : correct or Type I
Do not reject : correct or Type II
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 40 Example: Errors in a Medical Context Background: A medical test is carried out for a disease (HIV). Questions: What does claim? What are the implications of a Type I Error? What are the implications of a Type II Error? Which type of error is more worrisome?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 47 Sample Mean Standardizing to z
If is known, standardized follows
z (standard normal) distribution:
If is unknown and n is large enough
(20 or 30), then and
Can use z if is known or n is large.
What if is unknown and n is small?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 48 Sample mean standardizing to t
For unknown and n small,
t (like z ) centered at 0 since centered at t (like z ) symmetric and bell-shaped if normal t more spread than z (s.d.>1) [ s gives less info] t has “ n -1 degrees of freedom”(spread depends on n ) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 49 Inference About Mean Based on z or t
known standardized is z
(may use z if unknown but n large)
unknown standardized is t
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 50 Inference by Hand Based on z or t
z used if known or n large
t used if unknown and n small
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 51 z distribution (Review) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 52 t distribution (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 53
Example: Distribution of t vs. z
Background : For n =9, has 8 df. Question: How does P( t >2) compare to P( z >2)? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L25. 55
Example: Distribution of t vs. z
Background : For n =9, has 8 df. Response: P( t >2) between _____ and _____
