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M 4/3/06
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Classes
resume.
Quiz and Discussion (both graded)
on How to Lie With Statistics. If
you missed the book distribution before spring break, be sure to visit
Amazon.com or a public library so that you can read this classic before
today. It is an enjoyable, quick read, and although the examples and dollar
values are dated, the principles are as valid today as they were 50 years
ago.
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T 4/4/06
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HW due: Follow this
link (same as the first link in the “AP Exam Review” section) and work
free-response problem #4 from the 2005 Form B exam. Record the number of
minutes required for each part. For example, write a “6” next to (a) and a
“7” next to (b) if it takes you 6 minutes for the first part and 7 minutes
for the second part. Also indicate whether you worked the problem with or
without notes. Try working the problem at first without notes, but obviously,
if you cannot make any progress, you will need to break out your notes or
review book.
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W 4/5/06
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HW due:
1. If you did not do the problem Monday night, then follow yesterday’s
link and record your timings as instructed in yesterday’s calendar entry.
2. Grade your work for #4, 2005 Form B. Use the scoring rubric provided by
the College Board. Everyone who intends to take the AP exam is expected to do
this assignment. (Others will lose points if they fail to do it, but they
will not have the tinge of disappointment attached.)
3. Everyone should then redo the problem without notes, attempting to satisfy
a 100% score. Time limit on the redo: 13 minutes (19.5 for extended time).
If this assignment takes you more than an hour because you did not do any HW
on Monday night, then I apologize, but you are still expected to do it.
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Th 4/6/06
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HW due:
1. Read the Barron’s pages on the topic of “chi square goodness of fit,” pp.
327-331.
2. Optional: Visit me in Math Lab to acquire the Chi Square Deluxe (CSDELUXE)
program for your calculator.
3. Read through the solution (below) of a problem similar to the one posed in
class.
4. Find a die and roll it 35 times to generate your own data set. Perform a
statistical test to see whether there is any evidence of bias in the relative
frequencies. Show all of the PHA(S)TPC steps, including the terms that are
added up in the calculation of the c2 test
statistic. Follow the format of the example solution below.
Problem:
In 31 rolls of a die, we obtain the results shown below. Is there any
evidence of bias in the relative frequencies?
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Roll
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# of Occurrences
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1
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2
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2
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7
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3
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9
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4
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4
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5
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2
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6
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7
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Solution:
Parameters: Let p1 =
true probability of a 1, p2
= true probability of a 2, . . . , p6
= true probability of a 6
Hypotheses: H0: The pi values are each 1/6.
Ha: Not all of the pi values are 1/6. (In
other words, at least one probability is not equal to 1/6.)
Assumptions [from STAT TESTS
handout]:
SRS
of rolls that could be made ü
Cells
represent counts ü
All
expected counts are ³ 1 ü (obviously, since all expected counts are 31/6)
No
more than 20% of expected counts are < 5 ü (in fact, none of the expected counts are < 5)
Sketch of sampling distribution of c2,
assuming that H0 is
true:
Test statistic: c2 = S (obs. – exp.)2/exp. = (2 – 31/6)2/(31/6)
+ (7 – 31/6)2/(31/6) + (9 – 31/6)2/(31/6) + (4 – 31/6)2/(31/6)
+ (2 – 31/6)2/(31/6) + (7 – 31/6)2/(31/6) » 1.9409 + . 65054 + 2.8441 + .26344 + 1.9409 +
.65054 » 8.290.
P-value: Use df = # of bins – 1 = 6
– 1 = 5. From table on p.15 of the AP
Form B test booklet (or equivalent table in the Barron’s book), a c2 value
of 8.29 with 5 degrees of freedom produces a P value between .10 and .15, and obviously closer to .15. Or, use
calculator’s c2cdf
function to get P = .141. The
keystrokes are c2cdf(8.29042,99999,5)
ENTER. You do not ever need to show any work for this step. In fact, you
should not show work if your “work” consists of calculator notation. If you
are using CSDELUXE, finding the P-value
is a snap. Simply punch your expected and observed counts into L2
and L3, and the program will do all the calculations for you.
Conclusion: There is no evidence (c2 =
8.290, df = 5, P = .141) that the
true probabilities of rolling 1, 2, 3, 4, 5, or 6 differ from the
hypothesized uniform values of 1/6. The experimental results, while far from uniform,
can plausibly be explained by chance alone.
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F 4/7/06
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Big HW due (30 pts.): Using your M&M data from yesterday, determine
whether there is significant evidence (at the a = .05 level) of departure from the proportions of colors
claimed by the manufacturer. Show all steps (PHASTPC style), including the
details in the computation of the test statistic. Correctness counts,
although minor rounding errors will be ignored. You may work with classmates,
but there is no point in comparing answers, since each of you has a different
set of data.
Note to Paul J.: I was unable to duplicate the bug you found in class
yesterday. My suspicion is that a line or two had somehow been deleted from
your program. Please reload a fresh copy of Chi Square Deluxe (CSDELUXE) or do
the work manually.
Note to students who forgot to visit Math Lab after school to collect data:
You will need to buy a package of M&M’s in order to complete this
assignment. People who were excused Thursday for the entire day have an
automatic extension until Monday, but people who were excused late or who
took a cut will be expected to submit the assignment on time.
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M 4/10/06
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HW due: Read pp. 301-304, 331-338; write pp. 321-322 #1, 3.
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T 4/11/06
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HW due: Write pp. 343-346 #2, 7, 8; pp. 350-351 #2, 4.
Allow 3 minutes for each of the multiple-choice questions and 13 minutes for
each of the free-response questions (PHASTPC format). That is a total of 35
minutes.
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W 4/12/06
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Quest
(50 points) on Hypothesis Testing, Including All Types of c2 Tests. Questions may be drawn directly from the Barron’s book,
with possible modification. Other questions will come from the Must-Pass Quiz, especially the
starred questions. In your hypothesis testing, you must follow full PHASTPC
procedures, including the checking of assumptions. You must either memorize
the assumptions (as summarized on the TI-83 STAT TESTS Summary) or
somehow store them on your calculator. Everything on that sheet is fair game
for the quest, except for STAT TESTS E (linear regression t test).
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Th 4/13/06
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HW due:
1. Read the remainder of the textbook (inference for linear regression).
2. Write up any 2 problems of your choice that deal with the subject of inference
for regression. I would recommend doing, at a minimum, one hypothesis test
involving the LSRL slope and one confidence interval involving the LSRL
slope. Show your work.
Helpful hints:
- If a hypothesis test is required, the null
hypothesis would be that b (the “true slope”) is 0 or, equivalently, that
r (the “true linear correlation coefficient”) is 0. Simply use
STAT TESTS E and punch the buttons.
- Confidence intervals are a little harder. Start
by using STAT TESTS E as if you were running a test. This will produce
valuable statistics, namely b1
(which your calculator calls b)
and t. Knowing b1 and t, you can calculate the s.e. of
the slope in one of two ways: (a) by using the absurdly complicated
formula at the bottom of the first page of your formula sheet, or (b) by
using the much simpler formula b1/t that I asked you to replace it
with. Since m.o.e. is always (crit. value)(s.e.), you are now home free.
(The critical value is found in the t
table, using df = n – 2, where
n = # of ordered pairs in the
data set.)
In class: Discussion of Nepal
Vitamin A supplementation.
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F 4/14/06
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HW due:
1. Read the Must-Pass Quiz (all
100 questions) in order to familiarize yourself with its contents. There will
be an oral quiz today on the starred questions.
2. For the “shoe size as explained by height” data we collected in class
yesterday, compute a 95% confidence interval for the true slope. Show the
PHASTPC steps. (Of course, since this is a confidence interval problem, you
may omit H, S, and P, and instead of calculating a test statistic in the T
step you will be looking up a t*
value.) Here are the data points in case you have lost them:
{(69, 11), (67, 9.5), (71, 11), (77, 14), (76, 12), (76, 13), (71, 12), (73,
12), (72, 13),
(73, 12), (71, 10.5), (73, 13), (71, 12), (70, 10), (67, 10), (71, 10.5),
(72, 13), (77, 14)}
Correction: In class yesterday, I
incorrectly stated that what your calculator displays as s following a STAT TESTS E operation is the standard error of the
slope. In fact, the s that your
calculator produces is the numerator of the fraction in the ridiculously
complicated expression for the standard error of the slope (the expression
that appears at the bottom of the first page of your formula sheet). If you
divide s by the square root of the
sum of the squared deviations of xi
from the sample mean of the x
values, you will get the s.e. of the slope. (Of course, you would do that
only if you were a glutton for punishment. By far the easier way to find the
s.e. of the slope is to use the formula b1/t that we have discussed on several
occasions.)
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M 4/17/06
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HW due: Work a mixture of multiple-choice and free-response
problems in the review book. Keep a time log, and show your work. By the
amount of time you spend on this task, I will be able to gauge your
seriousness of purpose for the AP exam. The minimum requirement for full
credit is 20 minutes, which would allow you to do 3 multiple-choice questions
and a standard-length free-response question.
Oral Quiz (10 points): Prepare for an oral quiz on the Must-Pass Quiz (all 100 questions).
You will receive one question, chosen at random. If you fumble it, you will
be given a second question, and the higher of the two scores will count. Exception: If the question you receive
is a starred question, you must answer it correctly in order to earn any
points. If the first question you
receive is a starred question, there will be no second chance.
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T 4/18/06
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HW due: Same as yesterday, except a minimum of 35 minutes
this time.
Oral and Written Quizzes (10 points
each). We will finish yesterday’s oral quiz, and there will also be one
or two written questions for everyone.
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W 4/19/06
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HW due: Same as yesterday, with another 35 minutes of new
work. I will also plan to re-scan Monday’s assignment, as well as (possibly)
one or more assignments from last week.
Oral Quiz (conclusion).
In class: Review of hypothesis testing and confidence intervals, including
LSRL t procedures.
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Th 4/20/06
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Test on Hypothesis Testing and
Confidence Intervals.
Essentially, this is the same coverage as last week, except for the addition
of LSRL t procedures. Please look
at your quest (or check your mailbox for the
graded version of the first page) to see all the correct answers. Scoring
guidelines are provided under the quest
link.
Here is a comprehensive topic checklist to help you study.
- PHA(S)TPC
procedures
- Examples of proper wording of conclusions:
“There is strong evidence (t =
7.9, df = 6, P < .001) that
training increases the true mean French score.”
“There is no evidence ( z = –.179, P = .858) that the true proportions of diseased individuals
are different in the two populations.”
“We are 90% confident that the true mean melting point is 32.53 degrees ± 1.81 degrees.”
“We are 95% confident that the true difference between support for the
two candidates is 2.8 ± 1.5 percentage points.”
“We are 95% confident that the true difference between the mean
achievement test scores is in the interval (–1.3, 2.9).”
- How to convert a C.I. in the (lower bound,
upper bound) format to the estimate ± m.o.e. format, and vice versa
- Assumptions from TI-83 STAT TESTS Summary
- 1-sample t
procedures (STAT TESTS 2 for test, 8 for C.I.); note that these are also
used for matched pairs
- 2-sample t
procedures (STAT TESTS 4 for test, 0 for C.I.)
- 1-proportion z procedures (STAT TESTS 5 for test, A for C.I.)
- 2-proportion z procedures (STAT TESTS 6 for test, B for C.I.)
- c2
goodness-of-fit test (CSDELUXE only; not found on calculator)
- c2
2-way test for independence or homogeneity of proportions (CSDELUXE or
STAT TESTS C)
- LSRL t
test for slope or linear correlation coefficient (STAT TESTS E)
- LSRL slope C.I. (STAT TESTS E, followed by
manual computation of s.e. of b1
to get m.o.e.)
- Degrees of freedom for each of the above,
presented here for your studying convenience:
df = n – 1 for 1-sample t
df = (incredible mess) for 2-sample t;
let your calc. compute df for you!
Note: Older texts recommend using min(n1 – 1, n2 – 1) as a conservative value. The calc. result
is better.
df = ¥ for 1-prop. z
df = ¥ for 2-prop. z
df = # of bins – 1 for c2
goodness-of-fit
df = (# of rows – 1) · (# of cols. – 1) for c2
2-way
df = n – 2 for LSRL t procedures
- Sampling distribution
- Test statistic
- P-value
- a level
- Type I error
- Type II error
- Power
- Standard error (s.d. of a statistic in a sampling
distribution)
- Sampling error
- Critical value
- Margin of error (m.o.e. = crit. value · s.e.)
- Confidence level vs. confidence interval
- 1-tailed vs. 2-tailed
- How to avoid blunders involving the word
“probability” in C.I. problems
- How to recognize and deal with matched pairs
- Statistical significance
- Purpose of inferential statistics.
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F 4/21/06
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HW due: Free-response questions 1 and 2 from the 2005
AP exam.
In class: AP review (day 1 of 7). Discussion of Lower School science fair
projects and a decision tree to use when trying to decide what statistical
test is appropriate.
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M 4/24/06
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Phi
Beta Kappa Day (no school).
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T 4/25/06
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HW due: Please make a sincere effort to get caught up on
all previously assigned HW, and work free-response questions 3, 4, and 5 from
the 2005
AP exam. This is a total of 39 minutes of additional problems, which is
reasonable inasmuch as you have an extra-long weekend. Students who are not
essentially in compliance may receive the following from me:
1. A letter to deliver to Mr. Andreoli informing him that as much as you
would like to take the AP examination, you are not going to be ready to do it
because you are unwilling or unable to do homework in preparation.
2. A photocopy for your parents, and a return receipt for me to indicate that
they have received it.
If you have already canceled your plans to take the AP exam, then the letter
may be modified to say that because homework does not currently fit in with
your priorities, you are not particularly interested in earning a passing
grade for the fourth quarter, even though you are aware that homework accounts
for a significant fraction of your grade.
I apologize for being so strident, but it is time to get serious. If you are
able to get caught up over the long weekend, then you may ignore all the
unpleasantness above. You have my thanks, and all is forgiven.
In class: AP review (day 2 of 7). We will discuss problems 1-5 from the 2005
AP exam.
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W 4/26/06
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HW due: Question 6 from the 2005
AP exam. Please give it a solid effort (25 minutes) before peeking at the
scoring
guidelines.
In class: AP review (day 3 of 7). Today we will probably begin the Must-Pass Quiz in earnest, using
students in random order. You need to get at least 8 out of 10 correct in
order to pass (9 out of 11, 10 out of 12, etc. on subsequent trials).
Everyone must pass before the end of the year, hence the name. Starred
questions, if missed, cause an instant scratch. You may use your calculator
only during the questions having to do with assumptions. Printed or written
notes are not permitted.
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Th 4/27/06
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HW due: Questions 1, 2, and 3 from the 2005
AP Exam, Form B. Time limit: 35 minutes. If you cannot finish in 35
minutes, you may carry over the last part of question 3 until Friday. Please
do not peek unnecessarily at the scoring
guidelines.
In class: AP review (day 4 of 7) and Must-Pass
Quiz. You are encouraged to visit the tote
board to see how everyone is doing.
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F 4/28/06
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HW due: Question 6 from the 2005
AP Exam, Form B. Also finish the last portion of question 3 if you could
not finish it for yesterday.
In class: AP review (day 5 of 7).
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