Chapter 2
This chapter parallels the previous chapter (considering of
data collection issues, numerical and graphical summaries, and statistical
inference from empirical p-values) but for quantitative variables instead of
categorical variables. The themes of
considering the study design and exploring the data are reiterated to remind
students of their importance. Analyses
for quantitative data are a bit more complicated than for categorical data,
because no longer does one number suitably summarize a distribution by itself,
and we also need to focus on aspects such as shape, center, and spread in
describing these distributions. This
also leads to heavier use of Minitab for analyzing data (e.g., constructing
graphs and calculating numerical summaries) as well as for simulating
randomization distributions. If your
class does not meet regularly in a computer lab, you might want to consider
having students work through the initial study questions of several
investigations, saving up the Minitab analysis parts for when you can visit a
computer lab. Or if you do not have much
lab access, you could use computer projection to demonstrate the Minitab
analyses. Keep in mind that there are a
few menu differences if you are using Minitab 13 instead of Minitab 14 (see the
powerpoint slides for Day 8
of Stat 212). One thing you will want to
discuss with your students is the best way to save and import graphics for your
computer setting. Some things we’ve used
can be found here.
Section 2.1:
Summarizing Quantitative Data
Timing/Materials:
This section covers graphical and numerical summaries for
quantitative data, and these investigations will take several class
sessions. Students will be using Minitab
in Investigations 2.1.3 (oldfaithful.mtw), 2.1.5 (temps.mtw),
and 2.1.6 (fan03.mtw,
the ISCAM webpage also provides access to the most recent season’s data.). Instructions for replicating the output shown
in Investigation 2.1.2 (CloudSeeding.mtw) are
included as a Minitab Detour on p. 111.
Excel is used in Investigation 2.1.7 (housing.xls). Investigations 2.1.1 and 2.1.2 together should
take about 50-60 minutes. Investigations
2.1.3, 2.1.4, and 2.1.5 together should take another 60 minutes or so. Investigation 2.1.6 could take 40-50 minutes,
and Investigation 2.1.7 could take 50-60 minutes. You might consider assigning Investigation
2.1.6 as a lab assignment that students work on in pairs and complete the “write-up”
outside of class. Or you can expand on
the instructions for Practice Problem 2.1.7 as the lab-writeup
assignment. Investigation 2.1.7 explores
the mathematical properties of least squares estimation in this univariate case and can be skipped or moved outside of
class, perhaps as a “lab assignment.”
Investigation 2.1.1 is meant to provoke informal discussions
of anticipating variable behavior. You
may choose to wait until students have been introduced to histograms (in which
case it could also serve to practice terminology such as skewness). One goal is to help students get used to
having the variable along the horizontal axis with the vertical axis
representing the frequencies of observational units. Furthermore, we want to build student
intuition for how different variables might be expected to behave.
Students usually quickly identify graphs 1 and 6 as either the soda choice or the gender variable, the only
categorical variables. Reasonable
arguments can be made for either choice.
In fact, we try to resist telling students there is one “right answer”
(another habit of mind we want them to get into in this statistics class that
some students may not be expecting, as well as that writing coherent
explanations will be an expected skill in this class). We tell them we are more interested in their
justification than their final choice, but that we see how well they support
their answers and the consistency of their arguments. A clue could be given to remind students the
name of the course these 35 students were taking. This often leads students to select graph 1
as the gender variable, assuming the second bar represents a smaller proportion
of women in a class for engineers and scientists. Students usually pick graphs 2 and 3 (the two
skewed to the right graphs) as number of siblings and haircut cost. We do hope they will realize that graph 3,
with its gap in the second position and its longer right tail (encourage
students to try to put numerical values along the horizontal scale) is not
reasonable for number of siblings.
However the higher peak at $0 (free haircuts by friends) and the gap
between probably $5 and $10 does seem reasonable. (In fact, students often fail to think about
the graph possibly starting at 0.) We
also expect students to choose between height and guesses of age for graphs 4
and 5. Again, reasonable arguments could
be made for either, such as a more symmetric shape for height, as expected for
a biological characteristic? Or one
could argue for a skewed shape for height (especially if they felt the class
had a smaller proportion of women)?
Again, we evaluate their ability to justify the variable behavior, not
just their final choice. This
investigation also works well as a paired quiz but the habits of mind that this
investigation advocates were part of our motivation for moving it to first in
the section.
In Investigation 2.1.2 students are introduced to some of
the common graphical and numerical summaries used with quantitative data, while
still in the context of comparing the results of two experimental groups. We
present these fairly quickly, and we emphasize the subtler ideas of comparing distributions,
because we don't really want to pretend that these mathematically inclined
students have never seen a histogram or a median before! (Note: the lower whisker on
p. 108 extends to the value 1.)
After the Minitab detour (which they can verify outside of class), this
investigation concludes by having students transform the data. While not
involving calculus, transforming data is an idea that mathematically inclined
students find easier to handle than their more mathematically challenged peers. This part of the investigation can be skipped
but there are later investigations that assume they have seen this
transformation idea. You might also consider asking students to work on these
Minitab steps outside of class.
Practice 2.1.1 may seem straight-forward, but some students
struggle with it, and it does assess whether students understand how to
interpret boxplots.
Investigation 2.1.3 formally introduces measures of spread
and histograms. The data concern
observations of times between eruptions at
Investigation 2.1.4 asks students to think about how
measures of spread relate to histograms.
This is one of the rare times that we use hypothetical data, rather than
real data, because we have some very specifics points in mind. This is a “low-tech” activity that can really
catch some students in common misconceptions and you will definitely want to
give students time to think through (a)-(d) on their own first. The goal is to entice these students in a
safe environment to make some common errors like mistaking bumpiness and
variety for variability (as explained in the Discussion) so they can confront
their misconceptions head on. Our hope is that the resulting cognitive
dissonance will deepen the students' understanding of variability. It will be important to provide students
with immediate feedback on this investigation.
We encourage taking the time to have students calculate the interquartile ranges by hand as doing so for tallied data
appears to be nontrivial for them. This
is a very flexible investigation that you could plug into a 20-minute time slot
wherever it might fit in your schedule.
The actual numerical values for Practice 2.1.3 are below:
|
|
Jan |
Feb |
Mar |
Apr |
May |
Jun |
Jul |
Aug |
Sep |
Oct |
Nov |
Dec |
|
|
39 |
42 |
50 |
59 |
67 |
74 |
78 |
77 |
71 |
60 |
51 |
43 |
|
SF |
49 |
52 |
53 |
56 |
58 |
62 |
63 |
64 |
65 |
61 |
55 |
49 |
Investigation 2.1.5 aims to motivate the idea of
standardized scores for "comparing apples and oranges." While students may realize you are working
with a linear transformation of the data, we hope they will see the larger
message of trying to compare observations on different scales through
standardization. This idea of converting
to a common scale measuring the number of standard deviations away from the
mean will recur often. Practice 2.1.5
should help drive this point home. The
empirical rule is used to motivate an interpretation of standard deviation
(half-width of middle 68% of a symmetric distribution) that parallels their
understanding of IQR.
Investigation 2.1.6 gives students considerably more
practice in using Minitab to analyze data.
Students will probably need some help with questions (n)-(p) especially
if they are not baseball fans. These
questions can be addressed in class discussion where those that are baseball
fans can be the "experts" for the day. Still, we also want students to get into the
mental habit of playing detective as they explore data. We find Practice 2.1.7 helps transition the
data set to one that applies more directly to individual students. We encourage you to collect students' written
explanations (perhaps in pairs) to provide feedback on their report writing
skills (incorporating graphics and interpreting results in context). If this practice problem is treated more as a
lab assignment, you might consider a 20 point scale:
Defining MCI: 2 pts; Creating dotplots: 2 pts; Creating boxplots:
2 pts; Producing descriptive statistics: 2 pts; Discussion: 8 pts (shape,
center, spread, outliers); Removing one team and commenting on influence: 3
points.
Exploration 2.1 leads students to explore mathematical
properties of measures of center, and it also introduces the principle of least
squares in a univariate setting. As we mentioned above, this investigation can
be skipped or used as a group lab assignment.
Questions (a) and (b) motivate the need for some criterion by which to
compare point estimates, and questions (c)-(h) reveal that the mean serves as
the balance point of a distribution.
Beginning in (k), students use Excel to compare various other criteria,
principally for comparing the sum of absolute deviations and the sum of squared
deviations. Students who are not
familiar with Excel may need some help, particularly
with the “fill down” feature. Questions
(o) and (p) are meant to show students that the location(s) of the minimum SAD
value is not affected by the extremes but is affected by the middle. Students will be challenged to make a conjecture
in (q), but some students will realize that the median does the job. Questions (t)-(w) redo the analysis for the
sum of squared deviations, and in (x) students are asked to use calculus to
prove that the mean minimizes SSD. This
calculus derivation goes slowly for most students working on their own, so you
will want to decide whether to save time by leading them through that. Practice 2.1.8 extends the analysis to an odd
number of observations, where the SAD criterion now has a unique minimum at the
median. Practice 2.1.9 asks students
to create an example based on the resistance properties of these numerical
measures and is worth discussing even if Exploration 2.1 is not assigned.
Section 2.2:
Statistical Significance
Timing/Materials: Students are asked to conduct a
simulation using index cards in Investigation 2.2.1, followed-up by creating
and executing a Minitab macro. This
macro is used again in Investigation 2.2.2 and then modified to carry out an
analysis in Investigation 2.2.3. This
section might take 75-90 minutes.
This section again returns to the question of statistical
significance, as in Chapter 1, but now for a quantitative response
variable. Students will use shuffled
cards and then the computer to simulate a randomization distribution. However this time there will not be a
corresponding exact probability model (as there was with the hypergeometric distribution from Chapter 1), because we
need to consider not only the number of randomizations but also the value of
the difference in group means for each randomization, which is very
computationally intensive. We encourage
you to especially play up the context in Investigation 2.2.1, where students
can learn a powerful message about the effects of sleep deprivation. (It has
been shown that sleep deprivation impairs visual skills and thinking the next
day, and this study indicates that the negative effects persist 3 days
later.) The tactile simulation will
probably feel repetitive, so you may try to streamline it, but we have found
students still need some convincing on the process behind a randomization
test. It is also interesting to have
students examine medians as well as means.
In question (h) we again have students add their results to a dotplot on the board.
Students then use Minitab to replicate the randomization
process. They do this once by directly
typing the commands in Minitab (question j), where you might want to make sure
that they understand what each line is doing. (One common frustration is that
if students mis-type a Minitab command, they cannot
simply go back and edit it; they need to re-type or copy-and-paste the edited
correction at the most recent MTB> prompt.)
But then rather than have to copy-and-paste those line 1000 times, they
are then stepped through the creation of a Minitab macro to repeat their
commands and thus automate that process.
This is the first time students create and use a Minitab macro, in which
they provide Minitab with the relevant Session commands (instead of working
through the menus). Some students will
pick up these programming ideas very quickly, others
will need a fair bit of help. You may
want to pause a fair bit to make sure they understand
what is going on in Minitab. If a
majority of your students do not have programming background, you may want to
conduct a demonstration of the procedure first.
The two big issues are usually helping students save the file in a form
that is easily retrieved later and getting them into the habit of using (and
initializing!) a counter. We suggest
using a text editor (rather than a word processing program) for creating these
macro files so Minitab has less trouble with them. In fact, Notepad can be accessed under the
Tools menu in Minitab. It is also a nice
feature that this file can be kept open while the student runs the macro in
Minitab. In saving these files, you will
want to put double quotes around the file name.
This prevents the text editor from adding ".txt" to the file
name. The macro will still run perfectly
fine with a .txt extension but it is a little harder for Minitab to find the
file (it only automatically lists those files that have the .mtb extension - you would need to type *.txt in the File
name box first to be able to see and select the file if you don't use the .mtb extension). On
some computer systems, you also have to be rather careful in which directories
you save the file. You might want
students to get into the habit of saving their files onto personal disks or
onto the computer desktop for easier retrieval.
Remembering to initialize the counter (let k1=1 at the MTB> prompt)
before running the macro is the most common error that occurs; students also
need to be reminded that spelling and punctuation are crucial to the
functionality of the macro. We encourage
students to get into the habit of running the macro once to make sure it is
executing properly before they try to execute it 1000 times. These steps may require some trial and error
to smooth out the kinks.
In this investigation, you will want to be careful in
clarifying in which direction to carry out the subtraction (in fact for (k), we
suggest instead using let c6(k1)=mean(c5)-mean(c4)and
then in part (m), using let c8=(c6>=15.92), then consider the area to
right of +15.92 in the graph on p. 148).
Indicator variables, as in (m), will also be used extensively throughout
the text. We do show students the
results of generating all possible randomizations in (q) to convince them of
the intractability of this exact approach and to sow the seeds for later study
of the t distribution.
Investigations 2.2.2 and 2.2.3 provide further practice with
simulating a randomization test while focusing on two statistical issues: the
effect of within group variability on p-values (having already studied sample
size effects in Chapter 1) and the interpretation of p-values with
observational data. Question (b) of
Investigation 2.2.2 is a good example of how important we think it is to force
students to make predictions, regardless of whether or not their intuition
guides them well at this point; students struggle with this one, but we hope
that they better remember and understand the point (that more variability
within groups leads to less convincing evidence that the groups differ) through
having made a conjecture in the first place.
The subsequent practice problems are a bit more involved than most, so
you may want to incorporate them more into class and/or homework.