"Standard error of the (sample) mean" = s/sqrt(n)
SEM, SEXbar, etc.
Standard deviation of xbar, estimated from
the data.
Standard Error (SE) of an estimator/statistic= standard deviation of
the estimator/statistic, estimated from the data.
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-1 degrees of freedom.
We'll now repeat all the stuff from Chapter 6, only wherever there was
a z, we'll substitute a t.
It's a good idea to check
for at least approximate normality.
Confidence intervals: xbar +
t* (s/sqrt(n)) Choose t* from table D, using the n-1 row,
and confidence level C.
Special case of common pattern:
estimate + t* SEestimate
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 D,
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 milk 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
+ 1.833x268.45/sqrt(10)
4|889
4950 +
1.833x84.89,
or 4950
+155.6 bacteria/ml.
5|11
5|23
Test: H0 : mu = 4800
t = (4950 - 4800)/SEM
= 150/84.89 =
1.767
Ha : mu > 4800 (too contaminated)
t is between 1.383 and 1.833 (d.f. = 9)
P is between .10 and .05. Moderate evidence for Ha
--- --- --- --- --- ---
Robustness: (pp. 515-17) A procedure is
robust if the probability calculations required are insensitive
to violations of the assumptions made.
t-procedures are remarkably robust, more so for
larger n. Outliers are worse than nonnormality. Guidelines,
p. 516.
- - - - - - - - - - - - - - - - - - - - - - -
- -
If time: Boys
and guns newspaper clipping
The analysis of this is in the spirit of, though not exactly
the same as, " the bootstrap", p. 445-6
= = = = = = = = = = = = = = = = = =
HW Day 34: Read 7.1, to p. 517. We'll do matched
pairs (513-14) Monday after break, and use SPSS to do t-procedures.Skim
Power, p. 517-18 for the flavor. Then we'll do 518-23, Nonnormal;
continue with 7.2...
| Bring Monday:: your SPSS manual!
Hand in: p.522 One common mistake is to forget that the square root of n is already "in" the SE. If you're given the SE, don't divide by square root of n again! 7.1 critical values from table 7.3,4,5 one each for all the alternatives. Do all but the last parts. KEEP these three problems on a separate page; we will learn how to use SPSS to get exact P values and I will ask you then to complete these problems. 7.6 roommates Don't do the calculations, just answer the CI question. 7.14 corn prices 7.31 shrimp, s, CI 7.11, 12, 13 can openers--and transformations. To do these by hand: For 11a, do the stemplot, not the normal quantile plot. For 11 b, use xbar=23.56, s = 12.52 (I got them from the answers in the back of the book). Then you can do the rest as written. Part of Day 35 HW: Matched-pairs: the points on p. 514 are good. Often the "differences" will be more normal than the "befores" or the "afters" by themselves. 7.17, 18 piano lessons -->reasoning? For a, do a dot plot. For b, look in the answers for the mean and standard deviation, then go on. For the P-value, give the approximations from the table. |
Read, discuss
|
Optional
(more practice) 7.23-25 |
| Sievers home | Math251-Fall01/DayP34.htm | 11/19/01 |