1. Stat 31, Section 1, Last Time Sampling Distributions
Binomial Distribution
Binomial Probs
Normal Approx. to Binomial
Counts Scale vs. Proportion Scale

2. Important Announcement
2nd Midterm Date Changed,
from:
Tuesday, April 5, to
Tuesday, April 12.

3. Section 5.2: Distrib’n of Sample Means
Idea: Study Probability Structure of
Based on
Drawn independently
From same distribution,
Having Expected Value:
And Standard Deviation:

4. Expected Value of Sample Mean
How does relate to ?
Sample mean “has the same mean” as the original data.

5. Variance of Sample Mean
Study “spread” (i.e. quantify variation) of
Variance of Sample mean “reduced by ”

6. S. D. of Sample Mean
Since Standard Deviation is square root of Variance,
Take square roots to get:
S. D. of Sample mean “reduced by ”

7. Mean & S. D. of Sample Mean Summary: Averaging: Gives same centerpoint
Reduces variation by factor of
Called “Law of Averages, Part I”

8. Law of Averages, Part I Some consequences (worth noting):
To “double accuracy”, need 4 times as much data.
For 10 times accuracy”, need 100 times as much data.

9. Law of Averages, Part I
HW: (5.77, 4)

10. Distribution of Sample Mean
Now know center and spread, what about “shape of distribution”?
Case 1: If are indep.
CAN SHOW:
(knew these, news is “mound shape”)
Thus work with NORMDIST & NORMINV

11. Distribution of Sample Mean
Case 2: If are “almost anything”
STILL HAVE:
“approximately”

12. Distribution of Sample Mean
Remarks:
Mathematics: in terms of
Called “Law of Averages, Part II”
Also called “Central Limit Theorem”
Gives sense in which Normal Distribution is in the center
Hence name “Normal” (ostentatious?)

13. (any sum of small indep. Random pieces)
Law of Averages, Part II
More Remarks:
Thus we will work with NORMDIST & NORMINV a lot, for averages
This is why Normal Dist’n is good model for many different populations
(any sum of small indep. Random pieces)
Also explains Normal Approximation to the Binomial

14. Normal Approx. to Binomial
Explained by Law of Averages. II, since:
For X ~ Binomial (n.p)
Can represent X as:
Where:
Thus X is an average (rescaled sum), so Law of Averages gives Normal Dist’n

15. Law of Averages, Part II Nice Java Demo:
1 Dice (think n = 1): Average Dist’n is flat
2 Dice (n = 1): Average Dist’n is triangle …
5 Dice (n = 5): Looks quite “mound shaped”

16. Law of Averages, Part II Another cool one:
Create U shaped distribut’n with mouse
Simul. samples of size 2: non-Normal
Size n = 5: more normal
Size n = 10 or 25: mound shaped

17. Law of Averages, Part II Shows: Even starting from non-normal shape,
Class Example:
Shows:
Even starting from non-normal shape,
Averages become normal
More so for more averaging
SD smaller with more averaging ( )

18. Law of Averages, Part II
HW: , 5.33, 5.35, 5.39

19. And now for something completely different….
A statistics professor was describing sampling theory to his class, explaining how a sample can be studied and used to generalize to a population.
???

20. Chapter 6: Statistical Inference
Main Idea:
Form conclusions by
quantifying uncertainty
(will study several approaches,
first is…)

21. Section 6.1: Confidence Intervals
Background:
The sample mean, , is an “estimate” of the population mean,
How accurate?
(there is “variability”, how much?)

22. Confidence Intervals Recall the Sampling Distribution:
(maybe an approximation)

23. Confidence Intervals Thus understand error as:
How to explain to untrained consumers?
(who don’t know randomness,
distributions, normal curves)

24. Estimate +- margin of error
Confidence Intervals
Approach: present an interval
With endpoints:
Estimate +- margin of error
I.e.
reflecting variability
How to choose ?

25. Confidence Intervals Choice of “Confidence Interval radius”,
i.e. margin of error, :
Notes:
No Absolute Range (i.e. including “everything”) is available
From infinite tail of normal dist’n
So need to specify desired accuracy

26. Confidence Intervals Choice of “Confidence Interval radius”, :
Approach:
Choose a Confidence Level
Often 0.95
(e.g. FDA likes this number for
approving new drugs, and it
is a common standard for
publication in many fields)
And take margin of error to include that part of sampling distribution

27. Confidence Intervals E.g. For confidence level 0.95, want distribution
0.95 = Area
= margin of error

28. Confidence Intervals
Computation: Recall NORMINV takes areas (probs), and returns cutoffs
Issue: NORMINV works with lower areas
Note: lower tail
included

29. Confidence Intervals So adapt needed probs to lower areas….
When inner area = 0.95,
Right tail = 0.025
Shaded Area = 0.975
So need to compute:

30. Confidence Intervals Need to compute: Major problem: is unknown
But should answer depend on ?
“Accuracy” is only about spread
Not centerpoint
Need another view of the problem

31. Confidence Intervals Approach to unknown :
Recenter, i.e. look at dist’n
Key concept:
Centered at 0
Now can calculate as:

32. Confidence Intervals Computation of: Smaller Problem: Don’t know
Approach 1: Estimate with
Leads to complications
Will study later
Approach 2: Sometimes know

33. Confidence Intervals
138
139.1
113
132.5
140.7
109.7
118.9
134.8
109.6
127.3
115.6
130.4
130.2
111.7
105.5
E.g. Crop researchers plant 15 plots with a new variety of corn. The yields, in bushels per acre are:
Assume that = 10 bushels / acre

34. Confidence Intervals E.g. Find:
The 90% Confidence Interval for the mean value , for this type of corn.
The 95% Confidence Interval.
The 99% Confidence Interval.
How do the CIs change as the confidence level increases?
Solution, part 1 of:

35. Confidence Intervals An EXCEL shortcut: CONFIDENCE
Careful: parameter is:
2 tailed outer area
So for level = 0.90, = 0.10

36. Confidence Intervals
HW: 6.1, 6.3, 6.5