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Spring break.
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M 4/5/010
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HW due: Read pp. 689-700, 702-710,
713-723, and 737-740. Note: Skip
Activity 13.1, and ignore everything on p. 740 after the line that says,
“Standardized residual plot.” This is a total of 36 pages of reading spread
over 16 days, or an average of 2.25 pages per day. You can do this. Reading
notes are required, as always. This is the last textbook reading assignment
of the year. Hooray!
I can simplify your life somewhat by saying that everything you need to know
about checking assumptions for the LSRL t
test can be found on row E of this
handout. Note that this gives you only 3 things to check, instead of 4 as
indicated in the tan box at the top of p. 707.
In class: Double Quiz (20 pts.) on
the reading assignment. Both quizzes will be closed book, open notes. Any
handwritten notes you care to use will be acceptable.
Quiz I will consist of one
or two questions designed to test whether you actually did the assigned
reading, including all the examples. For example, anyone who did the reading
would surely remember a few salient facts about Example 3.1 (pp. 693-694) and
might be able to answer the following question correctly:
When wrestlers stand on their heads immediately before weigh-in, according to
the study cited in the textbook, the apparent weight loss (y) associated with wrestler weight
before headstand (x) exhibits which
of the following behaviors?
I. There is a positive
linear association between x and y.
II. There is a negative
linear association between x and y.
III. There is no statistically
significant linear association between x
and y, because y is essentially constant.
IV. The standard deviation of
LRSL residuals is approximately constant for any value of x.
(A) I only.
(B) II only.
(C) III only.
(D) I and IV only.
(E) III and IV only.
In order to answer this question correctly, you would need both (1) a vague
recollection of the example, based on your memory of an appropriate
upward-sloping linear model, implying that statement I is true, and (2)
knowledge of the entire chapter, not merely Example 3.1, since in the very
last line of p. 713 you are told that constant standard deviation of the
random deviations from the LSRL is a feature of appropriate models, and from
that you can conclude that statement IV is true. (Note: Your book uses
e on p. 713 as a synonym for residual.) Correct answer: D.
In other words, although the question is reasonable, it would require careful
reading of the entire 36 pages, and you might even need to read them a second
time in order to place everything into context in your mind.
- Will you need to do this in college? Yes.
- Is this level of reading required from you in
high school? Perhaps not often enough. However, I want you to succeed in
college.
Quiz II
will consist of a data set for which you need to determine (a) the df of the
LSRL t test, (b) the null
hypothesis for an appropriate test, and (c,d,e) the verification of the
assumptions. A completely worked example is given below.
Example problem: Mr. Hansen has noted that for the top 24 lunch skippers in
the Upper School during the first semester, there seems to be a slight linear
trend in their second semester skippiness. Here are the data:
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Semester 1
18.0
17.0
17.0
16.5
15.5
14.0
14.0
13.0
12.0
11.0
11.0
11.0
10.5
10.0
10.0
9.0
8.0
8.0
8.0
8.0
8.0
8.0
8.0
8.0
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Semester 2
5.0
4.0
5.5
9.0
9.0
4.0
0.0
2.0
5.5
4.0
6.0
8.0
5.0
4.0
5.0
2.0
1.0
2.0
6.3
6.5
5.0
3.0
9.0
1.0
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Answer Key
(a) Since there are 24 data points, df = n
− 2 = 22 for the LSRL t
test.
(b) H0:  , where  denotes the “true
slope” of the linear relationship between S1 and S2 lunch skips.
(c) Assumption 1 [adapted from tan box at top of p. 707; see also row E near the bottom of the STAT
TESTS summary handout]: Is the
true relationship linear? A scatterplot [omitted here to save space,
but you must show it] has r = .2097
and suggests a weak positive linear association between x and y, where x = S1 unexcused skips, y = S2 unexcused skips. The residual
plot [omitted here to save space, but you must show it] shows no obvious
patterns, suggesting that the linear fit is appropriate. We judge Assumption 1
to be satisfied, although some people might disagree. Proceed with caution.
[The scatterplot, residual plot, and r
value should all be shown for full credit, and you must state what it is
about the plots that makes you think the true relationship is linear. “No
pattern in residuals, hence good linear fit” or similar words must be written
so that the AP graders know that you know what you are talking about. Merely
sketching the plot and expecting the grader to fill in the gaps does not
work. You have to write down an actual justification of the evidence you are
presenting.]
(d) Assumption 2: Do the residuals
have a standard deviation that does not depend on x? To answer this question, we refer to the residual plot
found above and note that the vertical spread of the residuals does not
really seem to get larger or smaller as x
changes. This is a judgment call, of course. What we do not see is “flanging” or “flaring” in the residual plot [e.g.,
Figure 13.15(c) on p. 717], which would indicate a violation of the
assumption. We judge Assumption 2 to be satisfied.
(e) Assumption 3: Are the residuals
normally distributed? [Your book suggests on p. 714 that we should
try to calculate the standardized residuals using software but admits in the
second paragraph from the bottom of the page that these computations can be
“tedious.” Because we have no easy way to compute standardized residuals, we
will use plain residuals as suggested in Figure 3.14(b) on p. 716. This is
acceptable for the AP exam.] For the data set given, the LSRL is  , and the residuals are −.698, −1.541, −.0406,
3.5381, 3.6956, −1.068, −5.068, −2.911, .74656, −.596,
1.404, 3.404, .48269, −.4386, .56141, −2.281, −3.124, −2.124,
2.1763, 2.3763, .87626, −1.124, 4.8763, −3.124.
[Recall that the list called RESID is created automatically every time you
perform a STAT CALC 8 operation. To view the RESID list, if it is not already
shown on your STAT EDIT display, you can proceed as follows:
1. Press STAT EDIT.
2. Scroll all the way to the right until your cursor is on a blank line above
a column.
3. Press 2nd LIST.
4. On the NAMES menu, highlight RESID and press ENTER twice.
The RESID list can be checked for normality either by histogram, which is
acceptable for the AP exam, or by NQP, a normal quantile plot. The NQP
approach is better because it makes it easier to assess the skewness, if any.
If you have forgotten how to make a histogram or an NQP, you will need to
review your notes from the first semester. We did many examples of both.]
Since the NQP of the residuals [omitted here to save space, but you should
show it] shows a nearly perfect compliance with normality, we judge
Assumption 3 to be satisfied.
(f) This will not be on your quiz, but after doing all this work, wouldn’t
you like to know if there is statistically significant evidence of a positive
linear association between S1 and S2 lunch skips? Here is the PHA(S)TPC
writeup:
Let = “true slope” of
the linear relationship between S1 and S2 lunch skips, for the 24 skippiest
S1 students.
H0:
Ha: 
Assumptions for LSRL t test:
(1) Is the true relationship
linear? Yes, based on scatterplot, residual plot, and r = .2097 as described above. [All
three must be shown, and words such as “Lack of pattern in residual plot
suggests linear fit is valid” must be written so that the grader knows what
it is about the residual plot that makes you so certain.]
(2) Do the residuals have s.d. that
does not depend on x? Yes,
based on resid. plot [resid. plot must be shown]. The “spread” of residuals
does not “flange” or “flare out” as x
gets larger or smaller. [You need to write some words to that effect. Do not
expect the resid. plot to speak for itself.]
(3) Are the residuals normally
distributed? Yes, based on NQP of residuals [NQP of residuals should
be shown]. Straight line implies good normality.
Test statistic:  [by LinRegTTest]
P = .1627
Conclusion: There is no evidence (t = 1.0058, df = 22, P = .1627) that the true slope of the
regression line associating the 24 worst first-semester lunch skippers with
their second-semester results is positive. The slight upward trend seen in
the data can be plausibly explained by chance alone.
[There is no need to show the computation of  , the standard error of the slope, even though a
complicated formula for it is given on your AP formula sheet. That AP formula
is not useful, and you can cross it out. If you ever need to know , you can easily compute by a much easier
method. Since t is defined to be  , you can do a bit of algebra to obtain  . Therefore, since both b1
and t are readily produced by calc.
and/or displayed on a statistical software printout, you should be able to
find if you ever need to.
By the way, do not confuse with the value
called s that your calculator
displays when you perform STAT TESTS LinRegTTest. The calculator’s so-called s value is irrelevant for our purposes
here. If you are curious about what s
means in this context, ask me and I’ll explain how s is related to the standard deviation of the residuals, but that
is not anything you need to know for the AP exam.]
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T 4/6/010
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HW due: (1) Review your
reading assignment and freshen your reading notes so that the content is
fresh in your mind. (2) Finish your CFU, carefully correcting the quibble(s)
found among your answers. Note that there may be more than one error, even if
I mentioned only one in class. You may confer with your assigned partner, but
you may not copy anyone’s wording for the last question.
Graham and Ben are exempt from the second part of this assignment.
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W 4/7/010
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HW due: For the data set in
the 4/5 calendar entry, verify all three of the assumptions. Try to do this
without referring to my work. (You will learn more that way. You can and
should compare your answers when finished.)
Quiz will be over material similar
to Monday’s “disaster” quiz. I am thinking positive thoughts and hoping that
a majority of students do much better on this quiz. There will be a small
prize for the most improved student.
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Th 4/8/010
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HW due: Write #13.64 a-f
(omit part g) and start preparing for the Must-Pass Quiz by beginning a study
sheet of written notes. You will be graded on the quality of your study
sheet. For example, something that says “#19 no/yes” will not qualify for
credit. However, the following more thoughtful notes would qualify:
19. We know that r is unaffected by
choice of units, because r is
standardized (hence dimensionless). However, b0 and b1
must be expressed in the units of the problem and will change if the units
change. For example, if x is ocean
depth and y is pressure, b0 would be measured in y units (e.g., lbs./in.2),
while b1 would be
measured in 
units, namely something like pounds per square inch per foot. If these are
changed to metric units, then b0
and b1 would have to be
converted using the appropriate conversion factors (or recomputed from metric
source data, which might be just as easy).
Note: I understand that you may not
need notes for most of the questions. That is fine. Simply make notes for the
questions for which you need a little extra reinforcement. For today, only a
start is expected. Please keep a time log.
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F 4/9/010
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HW due: Continue working on
your notes for the Must-Pass Quiz.
Keep a time log. This assignment is due by 3:30 p.m. Thursday for those who
are attending today’s field trip.
In class: Work the following  two-way test. People
who missed class are expected to have this problem in their notes, worked.
A census of Schmilson H.S. has revealed the following categorical data:
Are hair color and eye color related in some way? Perform a suitable test and
analyze the results.
You are expected to have a worked version of this problem in your class notes
for Monday. Copying from somebody who was present is acceptable, since these
are class notes, not homework. To speed the computation of the contributions
to , we used an excerpt from the CSDELUXE program equivalent to the
following. Note that every square-bracketed entry, e.g., [A], must be entered
by using the MATRX function.
0 F
sum(dim[A]){1,0})R
sum(dim[A]){0,1})C
For(I,1,R,1)
For(J,1,C,1)
[A](I,J)O
[B](I,J)E
(O–E)2/E[C](I,J)
If E<5
Then
F+1F
End
End
End
Note: Before running the program,
you need to have matrix [A] populated with data, matrix [B] populated with
expected counts (either computed manually or produced automatically by the
STAT TESTS test procedure), and
matrix [C] defined with the same dimensions as [A] and [B].
After you have run the program above, F will contain the number of expected
counts that are less than 5. Remember, this count should not exceed 20% of
the number of entries in the matrix, which in this case would be 20% of 16,
or about 3.
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M 4/12/010
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HW due: Write #12.45,
12.46. Show all steps. Note: I will
allow all students to submit this assignment on Monday, even the students who
attended Friday’s field trip.
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T 4/13/010
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HW due: Continue reviewing
for your Must-Pass Quiz. Prepare
additional notes and/or work a selection of odd-numbered problems from
Chapters 12 and 13. Keep a time log in either case.
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W 4/14/010
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Test (100 pts.) on Chapters 12 and 13.
The useful CSDELUXE program is
available for TI-GRAPH LINK or TI CONNECT download, or you can visit me at
Math Lab to receive a copy by direct cable transfer. CSDELUXE covers both
g.o.f. and 2-way situations, and it even helps you with checking the expected
counts to see if they are large enough. You will want this program!
Today’s test will emphasize the following topic areas:
 g.o.f. tests- 2-way tests
(both for homogeneity of proportions and for independence)
- LSRL t
tests
- Confidence intervals for the LSRL slope.
You will be given a standard AP formula sheet to use during the test.
However, please note that one of the most useful formulas of all, namely  should be memorized,
since it is not on the AP formula sheet.
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Th 4/15/010
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Test (100 pts.) over same material as yesterday.
If you take both tests, only the higher of the two scores will count toward
your quarter average. If you wish to skip one day or the other in order to
use the time for studying for other classes, you must make special
arrangements with me in advance. Face-to-face is preferred. If you send me an
e-mail and do not receive a reply, you are not approved to skip a test, and you must appear for roll call
both Wednesday and Thursday as usual.
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F 4/16/010
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HW due: Write Wednesday’s free-response portion completely
(showing work). Set a timer for 25 minutes to make it challenging, and then
write everything beyond that point in a different color.
In class: The Must-Pass Quiz begins,
for real, with a 5-question written SRS for a quiz grade.
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M 4/19/010
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HW due: Show clear, written
evidence of work toward AP review (all students, even those who are not taking
the exam). A mixture of free-response problem solutions, written out as if
they were from a real exam, would be preferred. Multiple-choice questions, if
used as part of your review, must include some legible work. Keep a time log.
In class: The Must-Pass Quiz
continues with another quiz grade. Robbie gets an automatic 10 on this quiz,
because he passed last Friday. Congratulations, Robbie!
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T 4/20/010
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Diversity Day (no class).
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W 4/21/010
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HW due: Do parts (a) and
(b) of the following problem.
Problem: It is claimed that the
SAT I math exam (SATM) is scored and normed each year so that the population
approximately satisfies the the N(500,
100) distribution. [If you have forgotten what this notation means, then I
expect you to look it up.] Suppose that an SRS of 1648 SATM student papers
shows the following score distribution:
There are some anomalies. For example, by using normal probabilities, you can
easily determine that the expected number of scores above 750 would be
10.23355, and yet 20 people in the sample—almost double the expected
number—managed to score that high. [If you can’t determine 10.23355 as the
expected count for scores above 750, then e-mail me or call me immediately at
703-599-6624, since you will not be able to solve the problem.]
(a) Let the null hypothesis be that the population really is N(500, 100), and use a suitable  test with  = .05 to see if
there is any evidence that the claim is false. (Show all steps.) Note that
you cannot use the NQP approach, nor even the empirical rule, since the raw
data are not given.
(b) Analyze the contributions to to see where the
largest areas of possible concern may be.
In class: Survey critique.
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Th 4/22/010
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HW due: Adjust the mean and
s.d. in the null hypothesis for yesterday’s problem so that statistical
significance is assured. Show your work, as you did yesterday. Do both part (a)
and part (b).
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F 4/23/010
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HW due: Nothing new, but
previously assigned problems may be re-scanned, and accuracy is required for
problems that have been gone over in class. If there is anyone who does not
have homework today, I will send an e-mail to his advisor strongly
recommending that the student not sit for the AP Statistics exam. (Rationale:
If a student cannot do a few straightforward homework problems, I cannot
understand why such a student would want to sit in an exam room for more than
3 hours on a warm afternoon in May.)
In class: The Must-Pass Quiz
continues with another quiz grade.
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M 4/26/010
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Phi Beta Kappa Day (no
school).
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T 4/27/010
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The Must-Pass Quiz continues. Connor
and Eric go down in flames, for the moment.
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