MATH 251, Probability and Statistics I, Fall 2005, Mon. Oct. 31,  Day 28

HW Day 28  Read  5.2. For 5.2 homework, memorize the contents of the boxes, pp. 361-62. Please  skim through Weibull dist. (pp.367-8) Confidence intervals (6.1) tend to be hard to grasp conceptually; I urge you to read at least pp. 382-391 in preparation for the next class.
Hand in:  Sec. 5.2 
5.29 axle diameter (mean, s.d.)
5.31 abc axle diameter (cf. X and Xbar)
5.35  mean number of moths in a trap
5.41  airplane weight (They suggest changing it into a "mean" problem, but you can do it as a "sum" problem using the algebra of means and variances and noticing that if Xbar is normal, so is the sum of the  Xi's)
5.37, 5.39 Sheila's gestational diabetes
5.40 carpet flaws 
[5.39 and 5.40 both involve the computations needed for "significance testing".  The remaining step is this:  If our results are very unlikely, they call our assumption (the value of the mean) into question.]

Linear combinations of independent normal r.v.'s (as p.365-6) 
5.51 X+Y couples.
5.49 sum of 4 blocks
5.44 hole and shaft (Y-X)
5.64 screw-on caps break
5.46 difference of means--trustworthiness
5.47 difference of means--general		 Read, discuss 
  Sec.5.2
5.33 unbiased 5.65 p. 377 dye pH (Read for intro to "control limits", compare with 5.29 and 5.31.   This is "Quality Control")





5.45 diff. of means		 Optional 
Sec.5.2 (more practice) 
5.32 lightning strikes.  (Use algebra of means and variances for part a)

5.61 p. 377 car passengers.

Chaper 17, on your disk or downloadable, is all about issues like # 5.29, 31, 65--"Quality Control")


5.48 market gain: if results build on one another, usual mean may not be appropriate.
Homework questions?
Finish Sec. 5.2 (Central Limit Th. and friends) Day 27

"If X is normal then Xbar is normal"  is a special case of
A  linear combination of independent normal random variables is normal, (p.365-6) with the mean and s.d. you can calculate using the algebra of means and variances.
If X and Y are independent normal random variables,  then aX + bY is normal also.  Use "algebra" to find the mean and standard deviation.

Example:  Ann runs the mile in X minutes, where X is N(8, 2)
                Betty runs the mile in Y minutes, where Y is N(9, 3)
    Ann and Betty compete against each other a lot.  What proportion of the time does Betty beat Ann?  (Assume independence.)
       Want Prob. that Ann's time is longer than Betty's.  P(X > Y) = P(X - Y > 0)
                   Need distribution of X-Y.  It's Normal.  Mean =  µX - µY = 8 - 9 = -1
                              Variance = sigma2X  + (-1)2sigma2Y  = 22 + 32 = 4+9 = 13
                               S.d. = sqrt( 13) = 3.6   
   Back to P (X - Y > 0).  Standardize by subtracting the mean (-1)and dividing by the s.d. (3.6)
    P(X - Y > 0) = P(Z  > [(0-(-1))/3.6 ] ) = P(Z  > .28 ) = 1 -  .6103 = .3897, almost 40% of the time.

   Suppose each records her time for 4 days.  What's the probability that Betty's 4-day average  is faster than Ann's ?
     Want P( Xbar >Ybar ) = P( Xbar - Ybar > 0).   Need distribution of Xbar -Ybar.
                         The s.d. of Xbar is 2/2 =1  (since sqrt(4) = 2).  So Xbar is N(8, 1)
                          The s.d. of Ybar is 3/2 =1.5  (since sqrt(4) = 2).  So Ybar is N(9, 1.5)
                    Xbar -Ybar is Normal. Mean = 8 - 9 = -1.  Variance = 12 + (-1)2(1.5)2 = 1+2.25 = 3.25
                              S.d = sqrt(3.25) = 1.8
      P( Xbar - Ybar > 0) = P(Z  > [(0-(-1))/1.8 ] ) = P(Z  > .56 ) = 1 -  .7123 = .2877,  almost 30% of the time.

Statistics application:  Experiment: Two treatments; compare the mean results--what difference do we see?  If there really is an underlying difference (Ann is faster than Betty), are we sure of seeing it?  If there's no underlying difference, what's the chance of seeing something that looks like a difference?
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