Math 151 , Spring 2006, Day 21 Fri. Mar. 17 Hit reload ....After class.  Sample exam sol's linked

Exam 2 the Friday after break.  Covers Normal tables: raw<-->percentiles (Day 13 HW), thru  Monday Day 22 HW.
Sample exam handed out: solutions outside my door, + on reserve (soon) Scanned solutions: first attempt, as image files.  Email me if you have trouble with viewing them.  Page1,#1.  Page2,#2&3.  Page3,#4,5,&6.  Page4,#7a,b.  Page5,#7c,d&8. Page6,#9.  Page7,#10.
How much computational detail from part II?  You don't need to know the formula for the correlation coefficient, but you should be able to guess roughly the r from a scatterplot, and know and use the properties pp.121-2.You will need to know, among other things,  how to find b0 and b1 from the means, standard deviations, and r of the x-and y-values,  and to give the formula for the regression line, (like 17, p.154); and to graph the regression line on top of the scatterplot.  Also find by hand the value that the line predicts for a particular x.  You should be able to identify and calculate the residual value for a particular x-y point as its vertical distance from the line (negative if the point is below the line), and identify and understand potential influential points.  You should know  that the regression line goes through the point given by the two means, and that the  regression line "rises" r standard deviations in y for each standard deviation increase in x (pp. 137-8); also that the regression line of "weight" on "height" is not the same line as the regression line of "height" on "weight" . You should be able to describe verbally the meaning of R2 in the context of a data set.  "Extrapolation", other details from Ch. 9.
Day 21: Reading: D&V Ch 12, begin 13. Reading for Sampling: Ch. 11 pp.  216-7 The Step-by-step simulation effectively takes a random sample of size 3 from the 57 students.  The sampling processs is repeated 10 times.  The sampled individuals are only labeled as to whether they are Varsity or Not, but they could have been given names.  AS13 is good.
Hand in (All D&V p 238ff. unless otherwise noted)
1, 4, 6, 7, 8, 9 Do parts a,b,c,d, and e, and  f, and hand them all in. )

23 Sampling methods
21 Quality Control You may hand in the SPSS sample Wed. after break:
Use a cluster sample. (SPSS) Get the individuals  like this: 
In SPSS, type in values of a variable for case code, with values 61, 62,....80.  Get a sample of size3 & write down which cases are chosen.   Then  choose one from each case.  Consider them labeled from 1 to 12: enter those numbers into a variable.  For each case, take a sample of size 1 to decide which bottle from that case.   (Are we running a risk if we take the same (place) bottle from each case?)  Write down which bottles were chosen. 
AND when we've covered sampling from the Random Number Table (now!)(p. A-49) Do it again! (You'll get a different sample from your SPSS sample, of course. ) Use line 16 of the random number table, reading across, to first choose 3 cases from 61, 62,....80. Write down which cases are chosen.   Then  choose one bottle from each case like this.  Consider them labeled from 1 to 12:  For each case, take a sample of size 1 to decide which bottle from that case; keep reading the table where you left off.  Write down which bottles were chosen.   (I want everyone to start at line 16 so everyone will get the same sample and Fay can tell if you did it "right!") 

<>19 Accounting

11 Parent opinion I
15, 16 Phone and cell phone surveys

p. 267 #41 Security
p. 240 #20 Happy workers?  For e, use the Random Number Table (p. A-49) & read across Row 6. 
= = = = = = = = = = = = = = = 
Postpone Chapter 13   p257ff.
1,2,4,5,6,10,11,12,17,18   Do a: Decide if it is an observational study or an experiment.  If it was an observational = "investigative" study answer that b,c,d,e.  (We'll complete the experiment ones later; start the experiment ones on a separate page and keep it, answering the questions you can so far.) 
25 Wine
24 Full moon
 Hand in answers to these questions on the "Placebo Effect" articles (outside my door/on reserve ) Hand in Wed. after break: 
a) Give two examples of the placebo effect (from the article!)
b) What do researchers believe causes the placebo effect? 
c) In the separate article: "Pill will make you feel better...," what country was surveyed?

Read,
  to 
discuss 

p. 265 #29 Home-
coming

Postpone:
p. 239 #13 Wording
the survey
 

 Optional 

Homework questions? Day 20 
Recap:  Sample Chosen from a  Population
       (varies)             (fixed, but usually unknown)
Calculate
Numerical summary: Statistic (Latin) Parameter(Greek letter) (D&Vp227)
    Examples:           Sample mean xbar    Population mean mu (µ)
                       Sample st. dev. s    Pop. standard dev. sigma

The actual value of the Statistic will vary, depending on the particular sample.
"Sampling variability" = "sampling error"
The Statistic "estimates" the Parameter.  We hope it is close to the parameter.  If we choose simple random samples, we can understand the pattern of values the statistic can take.
Want sample to be representative of population, statistic to estimate paramater well, but variability happens...

Now: Simple Random Sample (SRS) of size n:   See Day 20 for details:
Sampling  frame.       Using SPSS to Sample. Get Handout.   Using Random Number Table to sample, see below.
Other sample designs:  Stratified random, Cluster, Multistage, Systematic
- - - - - - - - - - - - -
Sources of Bias in sampling: any systematic failure of a sample (or its method) to represent its population.  (E.g. sampling frame excludes a "different" part of population.)

Bad sampling designs:Not using randomness:

Other problems, even with good sampling design: Undercoverage:  of some or other group:  due to sampling frame, voluntary response bias, convenience sample, nonresponse....
Moral(s):  Design and plan to reduce potential biases
          Pretest for remaining problems   (A stitch in time saves 9; or the whole project)
          Report your sampling method and process in detail, so others may critique.

Using Random Number Table to sample (p. A-49)  Example: Ch. 11 pp.  216-7 The Step-by-step simulation effectively takes a random sample of size 3 from the 57 students.
    Every digit, every sequence of digits, is equally likely to be "next" in any direction. (Divisions into 5 is just for legibilty)
To use:  label everyone in the population with a number.
    Important:  Every labeling number needs the same number of digits.
    To label 9 people, use the labels 1,2,3,....9 (1-digit chunks)
    To label 15 people, use the labels 01, 02, ...10, 11, ...15 (2-digit chunks)
    To label 125 people, use the labels 001, 002, ... 124, 125 (3-digit chunks)
Pick a place (at random) in the table, start reading across in that size chunk.  Get n eligible numbers (discard repeats)
                    For example :   07511   88915   41267   16853   84569   79367 ..
From 9 people, a sample n = 5:   0,7, 5, 1, 1, 8, 8, 9, 1, 5, 4,     (sample is individuals 7, 5, 1, 8, 9)
From 15 people, a sample   07, 51, 18, 89, 15, 41, 26, 71, 68, 53, 84, 56, 97, 93, 67.... keep reading,
    go to next line (or back to top line) if you need more.  Individuals 7, 15,...are chosen using this line.
From 125 people, a sample 075, 118, 891, 541, 267, 168, 538, 456, 979, 367...keep reading.  Individuals 75, 118, ...

    Why the same number of digits in each label?  Each individual 3-digit chunk is as likely as any other 3-digit chunk.  But a 1- or 2-digit chunk is more likely than any 3-digit chunk. So 2 will come up more often than 12, but 02 will come up just as often as 12.
    Why across?  For consistency on HW, Start where I say and go across (so everyone who does it right gets the same answer.).  In practice, you can read up, down, backwards, as long as you decide beforehand, and don't change in the middle of choosing the sample.

= = = = = = = = = = = = = = = = = = = = = = = = = =
Start here Mon. after break:
D&V Ch13  Goal:  show cause-and-effect. Predictor-->Response
Observational Study:  Observe individuals; don't do anything to them; do not influence the responses.  Can indicate strength of relationship, differences, but not cause and effect.  (Often not with samples, but with selected group(s).)  Lurking variables?!? (Fisher:  Smokers smoke to soothe irritabilities that may cause cancer.)
         Retrospective:  gather data after the fact (observe that x% of men hospitalized with heart disease were/are smokers)
         Prospective:  choose individuals in advance. Measure them; or follow them, as events happen.  (Framingham Heart Study: 5,209 (2,873 women and 2,336 men) healthy residents between 30 and 60 years of age.  Followed from 1948 to now. A second-generation cohort recruited 1971, Minority group 1995  http://www.framingham.com/heart/)

Experiment: Impose treatments  on individuals, to see how the treatment influences  the response.
Compare treatments' effects.
Do something to:    "Experimental Units" = "Subjects"
   Treatment:  A Specific experimental condition.
   Factor: = Explanatory (Predictor) Variable we manipulate.
        Levels: Specific values of a factor that we set.
   Response variable(s)

E.g. 2 headache medications, in combination?
A two-factor experiment, each with 3 levels. 9 possible treatments.
    Factor A: Aspirin:  levels None, 500 mg, 1000 mg
    Factor B: Caffeine: levels None, 50 mg, 100 mg
Response variable: reported pain relief



Aspirin


None 500 mg 1000 mg

None Treatment 1 Treatment 2 Treatment 3
Caffeine 50 mg Treatment 4 Treatment 5 Treatment 6

100 mg Treatment 7 Treatment 8 Treatment 9

E.g. (Day 13, MRA-95-13 )Corn yield= response variable.  One Factor = Planting rate.  5 Levels=the rates.

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