1. Sampling Distribution of Sample Means Open Minitab
Stat 212 – Day 21
Sampling Distribution of Sample Means
Open Minitab

2. Last Time
If n is large, sampling distribution of sample proportions will be approximately normal with mean p and standard deviation
Test of Significance (H0: p=p0)
Use normal distribution to approximate p-value
Advantage: Also reports a test statistic
Confidence interval for p
Use normal distribution to approximate interval
Margin of error = half-width of interval (z*SE)
Some approximation methods are better than others

3. Interpretation of “confidence”
p. 4-41: I have calculated a 95% confidence interval for p, so there is a .95 probability that p is in the interval

4. Methods Simulation (empirical p-value)
If finite population, use hypergeometric distribution
If process or large population, use binomial distribution
Technical condition: population size > 20n
If large sample size, use normal distribution
Technical condition: np and n(1-p) > 10
Most flexible
Most common in practice right now
n > 10 and n(1- )>10
np0> 10 and n(1-p0)>10

5. Example: Bad News First
p=proportion of all St. Olaf students that prefer bad news first
H0: p=.5 (no preference in population)
Ha: p>.5 (preference for bad news first)
z = 3.0, p-value = .001, reject H0, significant pref
Exact p-value = .002
Level of significance
95% z interval for p: (.643, .957)
95% wilson interval: (.603, .914)
95% binomial interval: (.593, .932)

6. Last Time – Uncertainty in Statistics
Always the possibility are committing an error
Type I Error = rejecting a true null hypothesis
P(Type I Error) is controlled by level of significance
Type II Error = failing to reject a false null hypothesis
Power = probability of rejecting a false null hypothesis
Determine rejection region, how often will sample be in that rejection region if alternative is true
reject H0
reject H0

7. Power (Act 3-10)

8. Power (Act. 4-7, p. 4-32)

9. Power

10. Power

11. Power

12. Do the same thing for sample means!
Open milititamen.mtw and examine the distribution of chest measurements
Obs units? Variable? Type?
Describe distribution
Take random sample
Did everyone get sample mean?

13. Sampling distribution of sample mean
Run the macro militiamen.mtb
Describe the distribution. How compare to population?

14. Sampling distribution of sample mean
Open agemom.mtw
Describe the population
Re-initialize k1 and run macro again

15. Central Limit Theorem for Sample Mean
Sampling distribution of sample mean will be
Centered at m
Standard deviation s/
Normally distribution if population is normal or approximately normally distribution is sample size is large
Guideline: n>30

16. Confidence interval for m

17. Test of significant about m
H0: m=m0
Ha: m m < > or ≠
Observe “xbar”
Test statistic: (xbar – hypothesized mean) s/
Compare to t distribution with n-1 df

18. Activity 4-12

19. For Monday Practice problem 2,3, p. 4-52 Activity 4-12 p. 4-53 (a)-(i)
Use Internet Explorer
Uniform distribution
Activity 4-12 p (a)-(i)
Review p. 4-56
Start reading Ch. 5