| Hand in Monday . p. 434, 18.1 and 2, s<-->standard error p. 436, 18.3 Critical values: Use Table C and also the Excel t-procedures sheet; be sure your answers are consistent. p. 436, 18. 4 Critical values: Use Table C . For b, make a sketch. Note the decimal place is different in (a) and (b) p. 437 18.5 Critical values for CI. p. 437 18.7 Ancient air CI Make a dotplot or stemplot to examine the data. It will look somewhat skewed, but with so little data this kind of scatteredness can happen easily from a normal distribution. We should report that the skewness may make our CI only approximately accurate. Xbar = 59.5889% and s = 6.2553% are what you would get if you calculated from the data; use these to make your CI. Optional: Check with Excel t-procedures p. 453 18.29 absenteeism CI p. 455 18.36 a. Calcium and blood pressure CI (to use Table C where the degrees of freedom aren't given, go to the row with the lower degrees of freedom, here 50. You're giving up a little bit of sharpness rather than overstate your case.) Optional: To see how much difference the "correct" t* would give, use Excel t-procedures) p. 441 18.8 and 9 is it significant? Also, use Excel t-procedures to find the P-values more exactly. p. 432 18.25 read carefully. (one or more t-values was incorrectly computed.) p. 432 18.10 Ancient air test See note to 18.7, for mean and s.d. Optional: check with Excel t-procedures The rest will be assigned Monday. That's all . Using SPSS for one-sample procedures (with front page of Handout) ) A. Redo the example on the handout, getting the result of example 18.3 Dataset as SPSS file Dataset as text (.dat) file (If you import from the text file, remember to check that the Measure is Scale) p. 453, 18.27 Sharks (Use SPSS) Use SPSS to find the confidence interval, also to do the test, for the practice. p. 457, 18.41 Auto crankshafts (Use SPSS) = = = = = = = = = = = = = = = = = = = = = = Matched pairs, and robustness (by hand unless it says SPSS) p. 455, 18.37 measuring placebo effect Use Table C. You can check with the Excel t-procedures p. 446, 18.11and 12 newts healing. Find the differences by hand, and make a stemplot by hand. Use SPSS(back page of Handout) to do the test and CI. p. 448, 18.13 newts with outlier Use SPSS to do the tests. To eliminate the outlier, you can just delete that row from the data set. p. 450, 18.14 Reading scores Use Table C. Also, what IS the standard deviation? You can check with the Excel t-procedures . Also , you may find that the mean is (statistically) significantly below the basic level. Is the difference large enough to be important? (I don't know...) p. 455, 18.36b calcium/blood pressure conditions p. 454, 18.34 growing trees faster. Use SPSS. |
Read, to discuss |
Optional & & & Leftover problems from Day 30 & & & & & & & & These ideas are related to those in Ch. 15. You can get the answers visually by using the Statistical Significance Applet p. 290, 11.39 Pollutants in auto exhausts For 11.39: You might want to know L so that if you tested your 25 cars and found a high value of x-bar, you would be able to compare it with L; if it was greater than L, you would go back to the manufacturer and say "I believe you sold me a batch of bad cars, because the chances of getting an average emission level this high if the exhaust system is working properly is only 1 in 100. It is more reasonable to believe the exhaust system is not working, than that we "are" that 1 in 100 possibility." p. 290, 11.38 Glucose testing If we use this cutoff level L to say that people (with a mean of 4 tests) over L "have diabetes", then the chances of declaring that someone "has diabetes" when they really are OK (with mean 125mg/dl) is .05. .05 or 5% is the chance of a "false positive" using this protocol, when the real mean is 125. & & & & & & & & & & & & & & & & & & |
Look at shoeboxes, and simulations.
For the shoeboxes, the white numbers
(where the mean is really 20) rejected H0 : µ
= 20 (incorrectly) in favor of Ha : µ
> 20
at the alpha = .10 level in 3 of 18 samples (16.7%)
the yellow numbers (where the mean is really
bigger than 20--24 I think) rejected H0 : µ
= 20 (correctly!)
at the alpha = .10 level in 16 of 19 samples (84.2%)
Simulation: so far, with mean = 20,
reject H0 : µ = 20 (incorrectly)
at alpha = .10 in 31/250 or .124 of samples (close to
.10)
with mean = 24, reject H0
: µ = 20 (correctly!) at alpha = .10 in 181/250
or .724 of samples
See Day 39 for notes on old
problems, Ch. 18.
Ch. 18: Inference for population mean
(realistic)
What is the
significance
to Statistics of the Guinness Stout Bottle ?
Standard error of the (sample) mean = 
Standard deviation of xbar, estimated from the data.
"Standard
error of the mean": s/sqrt(n) SEM, SEXbar,
etc.
When you estimate the standard
deviation of a statistic, the resulting
estimate is called the "standard error" of
the
statistic.
t-distribution
family: like standard normal only slightly fatter in the
tails, slightly more spread.
Mean = 0. Symmetrical around 0.
t(k)
is the t distribution
with k degrees of freedom.
Comparison with normal (Excel
file)
Excel t-procedures
sheet
will find P's from t's.
Standardizing xbar with s instead of sigma results in
the one-sample t statistic, t-distribution with n-1degrees
of freedom.
Conditions for inference about a mean: (p. 434)
++ SRS (or reasonable facsimile)
++ Population is Normal. (Can relax
to symmetric, single-peaked unless n "very small")
"One-sample"
t- procedures:
SRS
of size n. Use Xbar
to estimate µ.
Confidence intervals: 
Choose t*
from table C, n-1
d.f., level C.
Significance tests:
State hypotheses
as in Ch. 15, find
t from data, by:
Calculating the one-sample
t-statistic, using the null hypothesis value of µ (call
it
µ0)
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___".
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,
xbar = 4950, s = 268.45 SEM = 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
t = (4950 - 4800)/SEM
= 150/84.89 =
1.767 |
Example: wax paper sandwich
bags:
Is the wax layer the same inside and out?
25 bags: measure (wax outside - wax inside)
for each. (pounds per square foot).
Differences: xbar
= .093, s = .723 n =
25
SEM = .723/5 = .1446
H0 : µ
= 0 (mean
difference
is
0)
t = (.093 - 0)/SEM
= .093/.1446
= .643.
Ha : µ
Not = 0 (there is a
difference)
t is less than .685 (d.f. = 24)
which is right-tail t* for probability .25
Because test is 2-sided, double the tail: .50. P value is greater
than .50.
No evidence for difference.
- - - - - - - - - - - - - - - - - - - - -
ROBUST procedures: a confidence
interval or significance test is called robust if the
confidence
level or P-value doesn't change very much when the assumptions of the
procedure
are violated. pp. 447-450.
Assumption: Population is Normal.
t-procedures are quite robust against
nonnormality.
But
sensitive
to outliers, bad skewness. Look at data. Need SRS!!
Details: n <15
t ok if data roughly symmetric, single peak, no outliers. Don't
use if skewed or outliers.
(How out is an outlier?)
n > 15 t ok unless there is strong skewness, or
outliers.
n > 40 or so: t ok even if there is skewness.
(Outliers?
I suggest trying with and without them, see what changes).
Matched-pairs data (differences) are often more normal in
shape than the separate variables ("oddness" is often the same for both
items in a pair, and disappears in subtraction. Another reason
why
this is a nice experimental design.
)
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