Math 151 , Fall 2002, Monday Day 37, Nov. 25 Hit reload to get most current versionAfter class

Question last Monday: there are no "numbers" attached to how big n, how close, how likely, in Law of Large Numbers:
"Take observations at random from a population with population mean µ. Then as the number of observations n increases, the sample mean xbar gets closer and closer to µ. "
   But Central Limit Theorem gives us a back door to such numbers:  for instance, if n = 100, then 95% of sequences of 100 randomly chosen numbers (SRS's of size 100) will have their xbars within about  2(sigma/10) of µ.  If we increase n to 400, then 95% of sequences of 400 numbers will have their xbars within about 2(sigma/20) of  µ.   We might be "unlucky" and get a sequence that stayed away from  µ longer, but eventually (not defined here) we should get close.
The Law of Large Numbers focuses on the behavior of Xbar as we look at larger and larger samples.  The Central Limit Theorem on the behavior of all possible xbars for a fixed sample size.

Sec 6.3, cont'd:   cautions and limitations:
Hawthorne effect and multiple test warning: Day 34

= = = = = = = = = = = = = = = = = = = = =
Chapter 7, Inference for Distributions (we'll do 7.1, 7.2, and the first segment, to p. 414, of 7.3)

Inference for means, using xbar from a SRS to make inference about µ:

	Large n  Sigma known          Sigma unknown		Small n  Sigma known          Sigma unknown
normal Population is  not normal	Xbar is normal;  find z using sigma	Xbar is normal;  find z using s.	Xbar is normal;  find z using sigma	Xbar is normal;  Find t using s
	Xbar is normal-ish (CLTh);  find z using sigma	Xbar is normal-ish (CLTh);  find z using s	Unrealistic. sigma's  only "good" for  normal pop's.	(See p. 381)  If you can't use t,  Find a statistician

t-distribution family:  like standard normal only slightly fatter in the tails.  Mean = 0. Symmetrical around 0.
    "Degrees of freedom" tell which member of the t family.  t(k) is the t distribution with k degrees of freedom.
 Comparison with normal (Excel file)
    Lower d.f.--fatter tails.  Higher d.f.--more like standard normal.
    Table C:  upper tail:  probability <--> "critical" t-value.

Start working on green box:
Assume Normal population .  Mean µ, s.d. sigma, both unknown.
Take SRS, size n, find xbar, find s (sample standard dev.)

Standard error of the (sample) mean = s/sqrt(n)    Standard deviation of xbar, estimated from the data.
  "Standard error of the mean":  s/sqrt(n) SEM, SE_Xbar, etc.  Just like sigma/sqrt(n), only s from data replaces sigma.
  When you estimate the standard deviation of a statistic,
                the resulting estimate is called the "standard error" of the statistic.

Standardizing xbar with s instead of sigma results in
t =    xbar -µ
         s/sqrt(n)       the one-sample t statistic
which has the t-distribution with n-1degrees of freedom.

We'll now repeat all the stuff from Chapter 6, only wherever there was a z, we'll substitute a t.
Here we go....
"One-sample" t- procedures: SRS of size n.  Use Xbar to estimate µ.
Substitute s for sigma in the standardizing formula. We get t instead of z, with n-1 degrees of freedom.
        It's a good idea to check for at least approximate normality.

Confidence intervals:  xbar + t* (s/sqrt(n))   Choose t* from table C, using the n-1 row, and confidence level C.
    Special case of common pattern:    estimate + t* SE_estimate

Significance tests:  State hypotheses as in Ch. 6, find t from data, by:
 Calculating the one-sample t-statistic, using the null hypothesis value of µ (call it µ₀)
t =    xbar -µ₀
         s/sqrt(n) Then proceed as if it were a "z", only using the (n-1) d.f. row in  table C,
to find P-values for the t*'s it's between, write "P-value is between ___ and___".
(Or use software which will find P-value exactly. )

Example: bacteria per milliliter in10 specimens of  raw milk from one producer.  Parameter: actual mean bacteria/ml.
         5370, 4890, 5100, 4500, 5260, 5150, 4900, 4760, 4700, 4870
4|5            n = 10,   xbar = 4950,  s = 268.45   SEM = 268.45/sqrt(10) =268.45/3.162=84.89.  deg. of freedom = 9
4|77          90% CI:  from t(9) in table,  t* = 1.833   CI is 4950+1.833^x268.45/sqrt(10)
4|889                                                                               4950 +1.833^x84.89, or  4950+155.6 bacteria/ml.
5|11                 If we had KNOWN Population sigma = 268.45, we'd have used z* = 1.645, gotten a narrower CI.
5|23                             (but we don't know sigma!)

                  Test:  H₀ : µ = 4800                                       t = (4950 - 4800)/SEM = 150/84.89 = 1.767
                              H_a : µ > 4800  (too contaminated)               t is between 1.383 and 1.833   (d.f. = 9)
                                                                                              P is between .10 and .05.  Some evidence for H_a
Monday: SPSS--Get Handout for 7.1.  Plug in above data and find P-value and CI.
    Analyze>Compare Means>One-Sample T Test:  Test value = 4800.
             P-value is labeled "Sig (2-tailed)"--divide by 2 for 1 tail (if observed is in correct direction)
   Analyze>Descriptive Statistics>Explore.  Statistics button, set Confidence level.
 

Activstats Ch's 18 (CI's) pp 1,2, 3 and Ch20 (tests) pp 1,2.  see Day 35. Also... - p. 20-3 (Power) Optional. For Next:  Matched-pairs experiments.  Moore pp. 374-78.  Activstats p. 20-4. (SPSS, activity 2, later) Then Two-sample procedures (Moore 7.2), Activstats Ch21, pp. 1-2.