Math 151 , Spring 2005, Day 37 Mon. May 2 Hit reload After Class

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

Day 37 Mon. May2: (Re)Read  Ch. 21 thru p. 392 (Activstats is good here too.) Then continue (Alpha levels) through 397.  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.)  Next: Chapter 23, Means.

Hand in (All D&V) hand in all A.  Use the T-table to decide these questions:  a)  Ho: p = .3 vs.  HA: p>.3.   z from p-hat is 2.12.  Is it significant at the .01 level? .05? .10?  b)  Ho: p = .3 vs.  HA: p not = .3.   z from p-hat is 2.12.  Is it significant at the .01 level? .05? .10?  c)  Ho: p = .3 vs.  HA: p>.3.   z from p-hat is 3.16.  Is it significant at the .01 level? .05? .10?  p. 387, #11 (use p.397--CI's & Tests) - - - - - - - - - - - - - - - -  Ch. 23, Inferences about means p. 447 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 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)

Read,   to  discuss

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

Making the decision to reject H_o
The T-table, bottom (z) row
CI can do two-sided test decision (approx.)   Day 34
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.  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