Stat 155 section 2 last time
This presentation is the property of its rightful owner.
Sponsored Links
1 / 51

Stat 155, Section 2, Last Time PowerPoint PPT Presentation


  • 115 Views
  • Uploaded on
  • Presentation posted in: General

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.

Download Presentation

Stat 155, Section 2, Last Time

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Stat 155 section 2 last time

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

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

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 results1

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 results2

Midterm I - Results

Too early

for letter

Grades:

Recall

Previous

scatterplot


Midterm i results3

Midterm I - Results

Interpretation of Scores:

  • 85 – 100 Very Pleased


Midterm i results4

Midterm I - Results

Interpretation of Scores:

  • 85 – 100 Very Pleased

  • 65 – 84 OK


Midterm i results5

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 results6

Midterm I - Results

Histogram

of Results:

Overall I’m

very pleased

relative to

other courses


Variance of random variables

Variance of Random Variables

Again consider discrete random variables:

Where distribution is summarized by a table,


Variance of random variables1

Variance of Random Variables

Again connect via frequentist approach:


Variance of random variables2

Variance of Random Variables

Again connect via frequentist approach:


Variance of random variables3

Variance of Random Variables

So define:

Variance of a distribution

As:

random variable


Variance of random variables4

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

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 variables5

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

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 variance1

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 variance2

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 variables6

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 variables7

Variance of Random Variables

HW:

C17: (cont.)

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


And now for something completely different

And now for something completely different

Recall

Distribution

of majors of

students in

this course:


And now for something completely different1

And now for something completely different

Couldn’t

Find

Any

Great

Jokes,

So…


And now for something completely different2

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

Chapter 5

Sampling Distributions

Idea: Extend probability tools to distributions we care about:

  • Counts in Political Polls

  • Measurement Error


Counts in political polls

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

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 distributions1

Binomial Distributions

HW on Binomial Assumptions:

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


Binomial distributions2

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

Binomial Probs in EXCEL

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

x

n

p


Binomial probs in excel1

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 excel2

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 excel3

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 excel4

Binomial Probs in EXCEL

HW:

5.3

5.4 (0.194)

Rework, using the Binomial Distribution: 4.52c,d


Binomial distribution

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 distribution1

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 distribution2

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 distribution3

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 distribution4

Binomial Distribution

HW on Mean and Variance:

5.5


Binomial distribution5

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 distribution6

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 distribution7

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”…


Binomial distribution8

Binomial Distribution

HW:

5.13

5.19


And now for something completely different3

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 different4

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 different5

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 different6

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 different7

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 different8

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 different9

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 different10

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”)


  • Login