MATH 251, Probability and Statistics I, Fall 2005, F Sept 2, Day 4 hit reload...After Class

Meet here Monday; in Mac 101 lab Wednesday for big SPSS intro.  Bring a disk or usb.
Unless otherwise noted, all assignments are in IPS
Day 4(Fri. Sept 7) Assigned: ( std. dev., linear transformations, densities)  Covers end of ch. 2, + pp. 66-9
Read for Wednesday's class, rest of 1.3 (Normal distributions), up to Normal quantile plots.
Use SPSS  Handout for computation of mean and std. dev, unless it says do it by hand.
Day 4 Hand in: 
Sec. 2.1  p.56ff. 
1.75 quintiles by hand. Method p. 45 middle
1.50 (Do xbar and s by hand.  Then put them in SPSS & do them.)
1.43 abc (you did the stemplot Day 2) Use SPSS for c (Table1.5)
1.77 (SPSS) Trimmed mean.  Do it like this: Load the guinea pig file (Table 1.8) into SPSS. Find the mean.  Then delete the highest 10% and lowest 10% of the observations (Click on the row, hit the Delete key). Find the mean of these = 10% trimmed mean.  Similarly find the 20% trimmed mean. (Median = 102.5, to do the comparisons. 
1.70 (SPSS) (computational accuracy)

1,72, 1.76 (Linear transformations)
A.  The mean August temperature in a certain Asian city is 25o C, with standard deviation 5o C.  What are these values in degrees Fahrenheit?  (f = 32 + 1.8 c)
p.98 1.141abc(ed scores "transformed to a standard scale")
   (hint for a: make 2 equations and solve for a and b)
Problem B below 

Postpone:
Sec. 1.3, p.84 1.80, 1.81, 1.82, (unif. density)
1.83 (mean, median, mode)

 Read, discuss
 

C. In problem B below, you need b > 0. 
Where does this come in to the computation--what would happen if you used a b that was negative?. 

 Optional 
 

Do 1.70 (computational accuracy) in Excel, if you're an Excel user.
B.  You have a data set x1, x2,... , xn,  which has mean xbar and standard deviation s.
You make a linear transformation xi*= a+b xi, on each data point.  (The book uses xnew  instead of x*, p. 54)
a can be + or - , but b should be positive.  (In practical terms, negative b would "flip" the data, reversing the order.)
Show that the mean xbar* of the transformed data set = a + b xbar,  (last sentence of p. 55)
and that the standard deviation of the transformed data set,  s* = bs .
(If you are not skilled at working with big sigmas, do it for n = 3 and write out all the sums with +'s.)
Do the proof by starting with the formula for the mean expressed in the xi*'s, e.g.  xbar*= (x1*+x1*+x3*)/3.
Plug in  xi* = a+b xi, and work the algebra to arrive at the desired expression (a+b xbar) involving the mean of the xi's.  Repeat for the  standard deviation formula.  Hint:  xbar* appears in the standard deviation expression:  substitute a +b xbar for it, since you already proved they were equal.

Email list: Math251@wells.edu    If you didn't get the welcome message, email lists@wells.edu
Cluster:  Tell everyone your name, even if you think they know it.
   Check for Homework questions? Especially 1.48, 1.64, "Read, to discuss" problems. Remaining #s on board.
HW:  PLEASE Label with Day #.  Please paperclip/staple.

We've been looking at SHAPE of distributions, and the ways irregularities can point us to knowledge about the data. (Living histograms.  As we Note p.49 middle:  Statistical [summary] measures and methods based on them are generally meaningful only for distributions of sufficiently regular shape. ... [Q]uickly resorting to fancy calculations is the mark of a statistical amateur.  Look, think, and choose your calculations selectively.

Note p. 53, fig. 1.20 shows a stemplot with negative numbers.
- - - - - - - - - - - - - - - - - - - - - - - - - - -
(Re)visit mean/median

Some other measures of middle:
    Mode (modal class), trimmed mean (throw away a % on each end), midrange (midway between min and max)

Spread, cont.
Standard deviation (goes with mean)
           Variance:  (almost) average of squared deviations from the mean.
                  (deviations sum to 0)
                 (Divide by (n-1) "degrees of freedom"--dimension of vector space spanning the deviations from the mean)
       s : Standard deviation  is the square root of the variance.  Formula p. 49-50.
                Computation:  I will require you to know how to do it by hand for up to 7 observations (use a table). Example.
             Physics: angular momemtum (spinning ice skater)
             Not so weird: High school geometry?
                Remember Pythagorean theorem: c2  = a2  + b2:
                hypotenuse of right triangle is also the square root of a sum of squares.
        Very sensitive to outliers (squared  deviations do it)
        >0 unless all observations are identical.
     Mean/standard deviation pair useful for symmetric, unimodal (one-humped), no outliers. ("Normal" dist.)
SPSS to find mean and s.d.   Handout

--------------------------------------------  -----------------------------------------
Linear transformations do not change the shape of a distribution :   A "good" measure of center or spread should "act naturally" if you change units of measurement by shifting (translating) or by stretching or shrinking (changing scale) .
     New x* = a + bx, for each observation.
Measures of spread are unaffected by the shifting! Only affected by the scale change.
Page 55 gives the rules explicitly.  Problem B has you prove them for mean and standard deviation.
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Monday: 1.3   Start with idea of Density function or curve: idealized histogram. Area = relative frequency.
Any curve that is above the x-axis and has area exactly 1 under it can be thought of as the idealization of some set of observations, and can be called a Density curve.  We carry over our terms for shape, and our summary measures.
Greek labels "mu" for mean and "sigma" for std. dev. of a Density.

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