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Stat 155, Section 2, Last Time. 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. Reading In Textbook.

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Stat 155, Section 2, Last Time
• 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
Reading In Textbook

Approximate Reading for Today’s Material:

Pages 277-286, 291-305

Approximate Reading for Next Class:

Pages 291-305, 334-351

Midterm I - Results

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
Midterm I - Results

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”
Midterm I - Results

Too early

for letter

Grades:

Recall

Previous

scatterplot

Midterm I - Results

Interpretation of Scores:

• 85 – 100 Very Pleased
Midterm I - Results

Interpretation of Scores:

• 85 – 100 Very Pleased
• 65 – 84 OK
Midterm I - Results

Interpretation of Scores:

• 85 – 100 Very Pleased
• 65 – 84 OK
• 0 – 64 Recommend Drop Course

(if not, let’s talk personally…)

Midterm I - Results

Histogram

of Results:

Overall I’m

very pleased

relative to

other courses

Variance of Random Variables

Again consider discrete random variables:

Where distribution is summarized by a table,

Variance of Random Variables

Again connect via frequentist approach:

Variance of Random Variables

Again connect via frequentist approach:

Variance of Random Variables

So define:

Variance of a distribution

As:

random variable

Variance of Random Variables

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)

Standard Deviation

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)

Variance of Random Variables

HW:

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

Properties of Variance
• 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)

Properties of Variance

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:

Properties of Variance

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)

Variance of Random Variables

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)

Variance of Random Variables

HW:

C17: (cont.)

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

And now for something completely different

Recall

Distribution

of majors of

students in

this course:

And now for something completely different

Couldn’t

Find

Any

Great

Jokes,

So…

And now for something completely different

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

Chapter 5

Sampling Distributions

Idea: Extend probability tools to distributions we care about:

• Counts in Political Polls
• Measurement Error
Counts in Political Polls

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.)

Binomial Distributions

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)
Binomial Distributions

HW on Binomial Assumptions:

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

Binomial Distributions

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

Binomial Probs in EXCEL

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

x

n

p

Binomial Probs in EXCEL

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

Cumulative:

P{X=x}: false

P{X<=x}: true

Binomial Probs in EXCEL

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

Binomial Probs in EXCEL

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…

Binomial Probs in EXCEL

HW:

5.3

5.4 (0.194)

Rework, using the Binomial Distribution: 4.52c,d

Binomial Distribution

“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)

Binomial Distribution

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”

Binomial Distribution

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)

Binomial Distribution

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

Binomial Distribution

HW on Mean and Variance:

5.5

Binomial Distribution

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”?

Binomial Distribution

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”?

Binomial Distribution

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”…

And now for something completely different

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

And now for something completely different

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

And now for something completely different

Main Ideas:

• For data base used in ratings
• Did careful analysis
• In an unbiased way
• Studied several aspects
• An interesting issue:

Who was “best”?

And now for something completely different

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”?
And now for something completely different

Who was “best”?

• Approach:

Consider all possible ratings

(i.e. all sets of weights)

• Which places can be #1?
• Which places can be “worst”?
And now for something completely different

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
And now for something completely different

Which places can be #1?

Which places can be “worst”?

Interesting fact:

Several cities on both lists!

And now for something completely different

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”)