Math 151 , Spring 2005, Day 38 Mon. Nov. 28 Hit reload  After class

Buffer against one low hour exam:
The final % exam grade minus 10 points will be substituted for the lowest hour exam grade, if it is higher.

Examples:		Ex1	Ex2	Ex3	final %	final -10
Student 1	Original	90	80	60	85	75, replaces lower 60
	Treated	90	80	75	85	ß These will be used.
Student 2	Original	90	80	70	75	65, lower than 70, don't replace.
	Treated	90	80	70	75
Student 3	Original	90	50	55	85	75, replaces lower 50
	Treated	90	75	55	85	<-These will be used

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

Day 38 : Continue  Ch. 21,  Lightly through Type I and II error,  Power.  Read What can go wrong p. 401 and the rest. (SPSS won't do proportion computations, but some other programs do; it's good to have an idea what you might see, p. 402.)  Read Chapter 23, Means.

Hand in (All D&V) Nothing to hand in: but READ! because I'll cover the chapter! Ch. 23, Inferences about means p. 447 DO: A)  Type your 4 numbers from the shoebox into SPSS, and use SPSS to find s, the sample standard deviation.  Save the data file, write down s.  Bring to class to add to sheet. Try these:  Hand in Friday: 1, 2 t-models:  Use the table in the text, p. A-53.  Check with Activstats if you want.  3, 4 more about t-models Postpone the rest: 5, 6 Cattle, Teachers  Nothing new except about mean instead of proportion.  7 Pulse rates Note the ME is half the total length of the CI  9 Normal temperature.(CI) Do this by hand now: we'll learn how to do it on SPSS "next"  13, 15 Hot dogs (CI) 21 Marriage (test) 25 Cars (test) B)  For your 4 numbers from the shoebox, find an 80% confidence interval for the mean of the shoebox population, by hand.

Read,   to  discuss

Optional  Postpone: Error type & power: p. 404, #7, #13

Get shoebox numbers, see above.  Shoebox will be outside my door.
Homework questions, Day 37
   Spent the class on HW:  Two sided tests and t-table: alphas.
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
Start here Wed.
Making the decision to reject H_o
The T-table, bottom (z) row
CI can do two-sided test decision (approx.)
Optional:  Later.  More about decisions in testing.
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.
       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 = 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