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Stat 155, Section 2, Last Time

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- Continuous Random Variables
- Probabilities modeled with areas

- Normal Curve
- Calculate in Excel: NORMDIST & NORMINV

- Means, i.e. Expected Values
- Useful for “average over many plays”

- Independence of Random Variables

Approximate Reading for Today’s Material:

Pages 277-286, 291-305

Approximate Reading for Next Class:

Pages 291-305, 334-351

Preliminary comments:

- Circled numbers are points taken off
- Total for each problem in brackets
- Points evenly divided among parts
- Page total in lower right corner
- Check those sum to total on front
- Overall score out of 100 points

Interpretation of Scores:

- Too early for letter grades
- These will change a lot:
- Some with good grades will relax
- Some with bad grades will wake up

- Don’t believe “A & C” average to “B”

Too early

for letter

Grades:

Recall

Previous

scatterplot

Interpretation of Scores:

- 85 – 100 Very Pleased

Interpretation of Scores:

- 85 – 100 Very Pleased
- 65 – 84 OK

Interpretation of Scores:

- 85 – 100 Very Pleased
- 65 – 84 OK
- 0 – 64 Recommend Drop Course
(if not, let’s talk personally…)

Histogram

of Results:

Overall I’m

very pleased

relative to

other courses

Again consider discrete random variables:

Where distribution is summarized by a table,

Again connect via frequentist approach:

Again connect via frequentist approach:

So define:

Variance of a distribution

As:

random variable

E. g. above game:

=(1/2)*5^2+(1/6)*1^2+(1/3)*8^2

Note: one acceptable Excel form, e.g. for exam (but there are many)

Recall standard deviation is square root of variance (same units as data)

E. g. above game:

Standard Deviation

=sqrt((1/2)*5^2+(1/6)*1^2+(1/3)*8^2)

HW:

C16: Find the variance and standard deviation of the distribution in 4.59. (0.752, 0.867)

- Linear transformation
I.e. “ignore shifts” var( ) = var ( )

(makes sense)

And scales come through squared

(recall s.d. on scale of data, var is square)

ii.For X and Y independent (important!)

I. e. Variance of sum is sum of variances

Here is where variance is “more natural” than standard deviation:

E. g. above game:

Recall “double the stakes”, gave same mean, as “play twice”, but seems different

Doubling:

Play twice, independently:

Note: playing more reduces uncertainty

(var quantifies this idea, will do more later)

HW:

C17: Suppose that the random variable X models winter daily maximum temperatures, and that X has mean 5o C and standard deviation 10o C. Let Y be the temp. in degrees Fahrenheit

(a) What is the mean of Y? (41oF)

Hint: Recall the conversion: C=(5/9)(F-32)

HW:

C17: (cont.)

(b) What is the standard deviation of Y? (18oF)

Recall

Distribution

of majors of

students in

this course:

Couldn’t

Find

Any

Great

Jokes,

So…

An Interesting and Relevant Issue:

- “Places Rated”
- Rankings Published by Several…
- We’ve been #1?
- Are we great ot what?
Will take a careful look later

Sampling Distributions

Idea: Extend probability tools to distributions we care about:

- Counts in Political Polls
- Measurement Error

Useful model: Binomial Distribution

Setting: n independent trials of an experiment with outcomes “Success” and “Failure”, with P{S} = p.

Say X = #S’s has a “Binomial(n,p) distribution”, and write “X ~ Bi(n,p)”

(parameters, like for Normal dist.)

Models much more than political polls:

E.g. Coin tossing

(recall saw “independence” was good)

E.g. Shooting free throws (in basketball)

- Is p always the same?
- Really independent? (turns out to be OK)

HW on Binomial Assumptions:

5.1, 5.2 (a. no, n?, b. yes, c. yes)

Could work out a formula for Binomial Probs,

but results are summarized in Excel function:

BINOMDIST

Example of Use:

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg19.xls

To compute P{X=x}, for X ~ Bi(n,p):

x

n

p

To compute P{X=x}, for X ~ Bi(n,p):

Cumulative:

P{X=x}: false

P{X<=x}: true

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg19.xls

Check this spreadsheet for details of other parts, and some important variations

Next time:

More slides on BINOMDIST,

And illustrate things like P{X < 3} = P{X <= 2}, etc.

Using a number line, and filled in dots…

HW:

5.3

5.4 (0.194)

Rework, using the Binomial Distribution: 4.52c,d

“Shape” of Binomial Distribution:

Use Probability Histogram

Just a bar graph, where heights are probabilities

Note: connected to previous histogram by frequentist view

(via histogram of repeated samples)

Study Distribution Shapes using Excel

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg20.xls

Part I: different p, note several ranges of p are shown

Part II: different n, note really “live in different areas”

A look under the hood

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg20.xls

Create probability histograms by:

- Create Column of xs (e.g. B9:B29)
- Create Probs (using BINOMDIST, C9:J29)
- Plot with Chart Wizard
Click Chart & Chart Wizard

Follow steps, check “series” carefully)

With some calculation, can show:

For X ~ Bi(n,p):

Mean: (# trials x P{S})

Variance:

S. D.:

Relate to (center & spread) of each histo:

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg20.xls

HW on Mean and Variance:

5.5

E.g.: Class HW on %Males at UNC:

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg17.xls Note Theoretical Means in E115:H115,

Compare to Sample Means in E110:H110:

Q1: Sample Mean smaller – course not representative

Q2: Sample Mean bigger – bias toward males

Q3: Sample Mean bigger – bias toward males

Q4: Sample Mean close

Which differences are “significant”?

E.g.: Class HW on %Males at UNC:

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg17.xls

Note Theoretical SDs in E116:H115,6

Compare to Sample SDs in E112:H112:

Q1: Sample SDs smaller – course population smaller

Q2: Sample SDs bigger – variety of doors (different p)

Q3: Sample SDs bigger – variety of choices (diff. p?)

Q4: Sample SDs close

Which differences are “significant”?

E.g.: Class HW on %Males at UNC:

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg17.xls

Probability Histograms (see 3rd column of plots),

Good view of above ideas (for samples):

Q1: mean too small, not enough spread

Q2: mean too big, too spread

Q3: mean too big, too spread

Q4: looks “about right”…

HW:

5.13

5.19

An Interesting and Relevant Issue:

- “Places Rated”
- Rankings Published by Several…
- We’ve been #1?
- Are we great ot what?
Will take a careful look now

Interesting Article:

Analysis of Data from the Places Rated Almanac

By: Richard A. Becker; Lorraine Denby; Robert McGill; Allan R. Wilks

Published in: The American Statistician, Vol. 41, No. 3. (Aug., 1987), pp. 169-186.

Hyperlink to JSTOR

Main Ideas:

- For data base used in ratings
- Did careful analysis
- In an unbiased way
- Studied several aspects
- An interesting issue:
Who was “best”?

Who was “best”?

- Data base had 8 factors
- How should we weight them?
- Evenly?
- Other choices?
- Just choose some?
(typical approach)

- Can we make our city “best”?

Who was “best”?

- Approach:
Consider all possible ratings

(i.e. all sets of weights)

- Which places can be #1?
- Which places can be “worst”?

Which places can be #1?

- 134 cities are “best”
- Including Raleigh Durham area
Which places can be “worst”?

- Even longer list here
- But Raleigh Durham not here

Which places can be #1?

Which places can be “worst”?

Interesting fact:

Several cities on both lists!

Some conclusions:

- Be very skeptical of such ratings?
- Ask: what happens if weights change?
- Think: what motivates the rater?
- Understand how other people can have different opinions
(Just different “personal weights”)