1. When we free ourselves of desire,
we will know serenity and freedom.

2. Inferences about Population Means
Sections
Estimation
Statistical tests: z test and t test
Sample size selection

3. Estimation To estimate a numerical summary in population (parameter):
Point estimator — the same numerical summary in sample; a statistic
Interval estimator — a “random” interval which includes the parameter most of time

4. Confidence Interval
Ideally, a short interval with high confidence interval is preferred.

6. Estimation for m Point estimator Confidence interval
Normality with known s or large sample: Z interval
Normality with unknown s: t interval

7. One Population

8. Sample Size for Estimating m
if s is unknown, use s from prior data or the upper bound of s.
Where E is the largest tolerable error.

9. The Logic of Hypothesis Test
“Assume Ho is a possible truth until proven false”
Analogical to
“Presumed innocent until proven guilty”
The logic of the US judicial system

10. Steps in Hypothesis Test
Set up the null (Ho) and alternative (Ha) hypotheses
Find an appropriate test statistic (T.S.)
Find the rejection region (R.R.)
Reject Ho if the observed test statistic falls in R.R.
Report the result in the context of the situation

11. Determine Ho and Ha The hypothesis with “=“ must be the Ho
The hypothesis we favor (called the research hypothesis) goes to Ha, if possible.
Eg. Example 5.7 (p.238)

12. Type II Error, or “ Error” Type I Error, or “ Error”
Types of Errors
H0 true
H1 true
Type II Error, or “ Error”
Good!
(Correct!)
we accept H0
Type I Error, or “ Error”
Good!
(Correct)
we reject H0
Type I error rate will be controlled at a given level, called significance level or a level

13. Z Test For normal populations or large samples (n > 30)
And the computed value of Z is denoted by Z*.

14. Types of Tests

15. Types of Tests

16. Types of Tests

17. Power of Test
Example 5.7 revisit (p. 240)

18. Sample Size for Testing m
The type I, II error rates are controlled
at a, b respectively and the maximum tolerable error is D:
One-tailed tests:
Two-tailed tests:

19. P-value (Observed Significant Level)
p-value is the probability of seeing what we observe as far as (or further) from Ho (in the direction of H1) given Ho is true; the smallest a level to reject Ho
p-value is computed by assuming Ho is true and then determining the probability of a result as extreme (or more extreme) as the observed test statistic in the direction of the H1.
The smaller p-value is, the less likely that what we observe will occur under the assumption Ho is true.
Smaller p-value means stronger evidence against Ho.

20. Computing the p-Value for the Z-Test

21. Computing the p-Value for the Z-Test

22. Computing the p-Value for the Z-Test
P-value = P(|Z| > |z*| )= 2 x P(Z > |z*|)

23. Hypothesis Test using p-Value
Set up the null (Ho) and alternative (H1) hypotheses
Find an appropriate test statistic (T.S.)
Find the p-value
Reject Ho if the p-value < a
Report the result in the context of the situation

24. Example 5.7 (Page 238)
Redo it using the p-value way

25. t Test For normal populations with unknown s
t = the same formula for Z
but replacing s by s
Eg. Revisit Example 5.7

26. One Population

27. Inferences about mean when “beyond the scope”
When population is nonnormal and n is small, how to do inferences about m:
1). Use Bootstrap methods to simulate the sampling distribution of t test statistic
2). Use the simulated distribution to find an (approximate) C.I. and p-value

28. Introduction to Bootstrap Methods
How to simulate the sampling distribution of a given statistic, say t, based on a given sample of size n:
Pretend the original sample is the entire population
Select a random sample of size n from the original sample (now the population) with replacement ; this is called a bootstrap sample
Calculate the t value of the bootstrap sample, t*
Repeat steps 2, 3 many times, 1000 or more, say B times. Use the obtained t* values to obtain an approximation to the sampling distribution
Minitab steps for obtaining bootstrap samples (p. 264)
Example 5.18, 5.19 (p )