Math 151 , Spring 2006, Day 38 Wed. May 3 Hit reload  after class

This is to encourage those who have had trouble to try to put it together for the final.

Homework questions, Day 36
  "Statistically significant" result, and  "alpha"  "significance level."   Rejecting  H_o . Cautions.  Notes Day 34
   Making the decision to reject H_o
   The T-table, bottom (z) row, to check significance at predetermined alpha levels Day 36
   CI can do two-sided test decision (approx.)
   More about decisions in testing.  See also inclass remarks, Day 35

t- family:  like standard normal only slightly fatter in the tails.  Mean = 0. Symmetrical around 0.
    "Degrees of freedom" (d.f.) 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 T p. A53
Start here Fri:
Inference with sample MEANS(Ch. 23)
Use y-bar to estimate unknown population mean µ:
Make Confidence Intervals and Tests, just as before...(almost)
  "Sampling distribution of the mean" (all possible y-bars from random samples) is Normal, with center at the unknown population mean µ.  If we knew the standard deviation sigma of the population, we could do tests and CI's just like for proportions:
 For CI,
For test of H_o = µ_o, Calculate z and find the P-value in the tail(s):
BUT we almost never know sigma!
So we substitute SE for SD--use the sample standard deviation in place of sigma.  This adds "slop"--more variability-- to our estimates.  Luckily we know exactly how much, if the population is (nearly) normal:
follows the "Student's t"  model.
t- 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 T p. A53: "One tail probability" (upper tail):  probability <--> "critical" t-value.
          (Activstats Reference has a "full" t-table, like the normal table, but with upper tails.)
is the one-sample t statistic
which has the t-distribution with n-1degrees of freedom.

We'll now repeat all the stuff from Part V, only wherever there was a z, we'll substitute a t.
Here we go....
"One-sample" t- procedures: SRS of size n.  Use Y-bar to estimate µ.
Substitute s for sigma in the standardizing formula. We get t instead of z, with n-1 degrees of freedom.
        You should check for at least approximate normality in the data set.  (see p. 435)

Confidence intervals: 
   Choose t* from Table T p. A53, using the n-1 row, and confidence level C.
    Special case of common patterns:    estimate + t* SE(estimate), or  estimate + z* SE(estimate)
Significance tests:  State hypotheses in terms of µ, find t from data, by:

 Calculating the one-sample t-statistic, using the null hypothesis value of µ (call it µ_o)
Then proceed as if it were a "z", only using the (n-1) d.f. row in  Table T p. A53,
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 in 10 specimens of  raw milk from one producer.
  Parameter: actual mean bacteria/ml. 
  Data: 5370, 4890, 5100, 4500, 5260, 5150, 4900, 4760, 4700, 4870

4|5  4|77 4|889  5|11  5|23

n = 10,   ybar = 4950, s = 268.45   SE = 268.45/sqrt(10) =268.45/3.162=84.89.  deg. of freedom = 10-1=9 90% CI:  from t(9) in table, t* = 1.833   CI is 4950+1.833x268.45/sqrt(10)                                                      4950 +1.833x84.89, or  4950+155.6 bacteria/ml. If we had KNOWN Population sigma = 268.45,    we'd have used z* = 1.645, gotten a narrower CI.   (but we don't know sigma!) Test:  H0 : µ = 4800  (OK)             t = (4950 - 4800)/SE = 150/84.89 = 1.767           HA : µ > 4800                        t is between 1.383 and 1.833   (d.f. = 9)              (too contaminated)            P is between .10 and .05.  Some evidence for HA (If the test had been 2-sided, P would be between .20 and .10)