S T A T I S T I C S
TEACHER’S GUIDE
Chapter V:
Scoring the Free-Response Questions
Copyright © 2002 the College Entrance Examination Board. All rights reserved
Chapter V
Scoring the Free-Response Questions
The short-answer questions and investigative task on the AP Statistics Exam are scored “holistically”; that
is, each response is evaluated as a complete package. With holistic scoring, a judgment is made about the
overall quality of the student’s response, as opposed to “analytic” scoring, wherein the necessary components of a student response are specified in advance and each component is given a value counting toward
the overall score. For example, suppose a student is to solve the quadratic equation x 2 2 x ! 7 . An
analytic scoring rubric might allocate a total of four points in the following manner.
Score one point for rewriting the equation as x 2 2 x 7 ! 0 .
Score one point for correctly choosing the quadratic formula.
Score one point for correct substitution of the coefficients into the formula.
Score one point for the correct answer.
An analytic scoring method is well suited for evaluating a response to a focused question with few
response alternatives (Nitko 2001, 194-195). Holistic scoring, in contrast, is well suited for questions
wherein a student is required to synthesize information and respond at least partly in written paragraphs.
For example, an open-ended question may present a “real life” experiment with resulting data and ask the
student not only to analyze the data but also to comment on how the experimental protocol might be
improved. Opinions on improvement might focus on improving the sampling method, controlling confounding variables, or seeking more power through increasing the sample size. Indeed, the student’s
justification of his or her experimental improvement could even depend on relevant contextual knowledge
brought to the exam from the real world! In this context, holistic scoring represents a recognition not only
of multiple reasonable approaches to a statistical analysis, but a statement about statistical synergy—a
quality student response is more than just the sum of its parts.
The Strategy of Holistic Scoring for the Free-Response Questions
One method of implementing holistic scoring is to decide in advance the number of categories into
which the student answers will be sorted. These may be labeled in any one of a number of ways: A, B, C,
D, and F; distinguished, proficient, and novice; or simple numbers, 5, 4, 3, 2, and 1. The AP Statistics
scoring rubric for each free-response question on the AP Exam has five categories numerically scored on a
0–4 scale. Each of these categories represents a level of quality in the student response. These levels of
quality are defined on two dimensions: statistical knowledge and communication. The specific rubrics for
each question are tied to a general template, which represents the descriptions of the quality levels as
envisioned by the AP Statistics Development Committee. This general template is given in the following
table, “A Guide to Scoring Free-Response Statistics Questions.”
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Scoring the Free-Response Questions
A Guide To Scoring Free-Response Statistics Questions:
The Category Descriptors
Score
Descriptors
Statistical Knowledge
Identification of the important
components of the problem
Communication
Explanation of what was done and
why, along with a statement of
conclusions drawn
Demonstration of the statistical
concepts and techniques that result in
a correct solution of the problem
62
4
Complete
• Shows complete understanding of the
problem’s statistical components
• Synthesizes a correct relationship
among these components, perhaps
with novelty and creativity
• Uses appropriate and correctly
executed statistical techniques
• May have minor arithmetic errors,
but answers are still reasonable
• Provides a clear, organized, and
complete explanation, using correct
terminology, of what was done and
why
• States appropriate assumptions and
caveats
• Uses diagrams or plots when
appropriate to aid in describing the
solution
• States an appropriate and complete
conclusion
3
Substantial
• Shows substantial understanding of the
problem’s statistical components
• Synthesizes a relationship among these
components, perhaps with minor gaps
• Uses appropriate statistical techniques
• May have arithmetic errors, but answers
are still reasonable
• Provides a clear but not perfectly
organized explanation, using correct
terminology, of what was done and
why, but explanation may be
slightly incomplete
• May miss necessary assumptions or
caveats
• Uses diagrams or plots when
appropriate to aid in describing the
solution
• States a conclusion that follows
from the analysis but may be
somewhat incomplete
Scoring the Free-Response Questions
Score
Descriptors
Statistical Knowledge
Communication
2
Developing
• Shows some understanding of the
problem’s statistical components
• Shows little in the way of a
relationship among these components
• Uses some appropriate statistical
techniques, but misses or misuses
others
• May have arithmetic errors that result
in unreasonable answers
• Provides some explanation of what
was done, but explanation may be
vague and difficult to interpret and
terminology may be somewhat
inappropriate
• Uses diagrams in an incomplete or
ineffective way, or diagrams may be
missing
• States a conclusion that is
incomplete
1
Minimal
• Shows limited understanding of the
problem’s statistical components by
failing to identify important
components
• Shows little ability to organize a
solution and may use irrelevant
information
• Misuses or fails to use appropriate
statistical techniques
• Has arithmetic errors that result in
unreasonable answers
• Provides minimal or unclear
explanation of what was done or
why it was done, and explanation
may not match the presented
solution
• Fails to use diagrams or plots, or uses
them incorrectly
• States an incorrect conclusion, or
fails to state a conclusion
0
• Shows little to no understanding of
statistical components
• Provides no explanation of a
legitimate strategy
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Scoring the Free-Response Questions
The AP Statistics Exam consists of two parts; the first is comprised of multiple-choice questions, the
second of free-response questions. The free-response questions and the holistic evaluation of the students’
responses to them reflect the practice of modern statistics and have implications for the instructional
strategies used when preparing students for the exam. The Topic Outline on page 28 delineates the skills
and concepts of the AP Statistics course. A student who understands the terms, correctly applies the
formulas, and can provide good answers to the “stock” questions in elementary statistics still may not be as
well prepared as possible for the open-ended questions. Why might this be?
The General Goals of the AP Statistics Course
The general goals of the course, consistent with the course philosophy, are more about the relations and
connections among the topics in the Topic Outline than about the topics alone. Instructional time should
encourage acquisition by students of an effective network of relationships that binds sets of topics into
cognitive wholes. The short-term benefit of such an instructional strategy is that students will be better
able to function when confronted with context-rich open-ended questions. The long-term benefits may be
divided into two classes. First, the instructional approach should go beyond “topics to be covered” and
synthesize relations among the topics to provide a solid foundation for further academic study in statistics.
Students preparing to study more advanced mathematics in college have a foundation in algebra and
geometry constructed throughout their high school careers. For many students, the analogous foundation
in statistics will be provided in the AP Statistics course. Statistics, like mathematics, is a discipline rich in
connections between parts that at first blush may not be readily seen as connected. A solid instruction in
statistics will help students see these connections. A second long-term benefit is the preparation of students
to address and solve problems of a statistical nature long after the statistics class has finished. Cognitive
psychology tells us that a relatively disconnected set of topics will be forgotten as time passes. A set of
topics cohesively organized into larger schema will be remembered longer and are more likely to be applied
in future problem-solving situations. The building blocks of such a cohesive framework are presented here.
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Scoring the Free-Response Questions
1. The Collection of Data
Students should understand that valid statistical analyses of data depend critically on the method of data
collection. Measures must be related to the variables of interest in the study. While this is generally not a
problem in the physical sciences, it is especially important, for example, when designing survey questions
in the social sciences. A survey consisting of vague or poorly worded questions will be difficult to interpret.
Meaningful generalizations from samples to populations can be made only if based on random sampling;
students should appreciate the importance of a good sampling plan. Also, they should be aware of the
differences between observation and experimentation, specifically with respect to the possibility of tentatively identifying a cause-and-effect relationship between variables. In general, students should be able to
construct a sampling plan, design an experiment, and interpret the results in light of these considerations,
as well as analyze and interpret a study with 20/20 hindsight, recognizing inevitable limitations and
offering constructive criticism.
2. The Representation of Data
Numeric and graphical representation of data is the starting point for both descriptive and inferential
statistical analysis. Whether data results from observations of individual subjects, or possibly from a simulation, the variability, shape of the distribution, and unusual or interesting features of data are of fundamental importance. In addition to providing information about the distribution of data values, simple
pictures of data can also uncover measurement errors, provide reality checks on assumptions about populations, and suggest avenues for future analysis. Students should be able to represent data in a variety of
forms and base sound statistical arguments on their representations.
3. Probability Is the Basic Language of Statistics
Probability, the mathematics of chance and variability, provides both foundational ideas like events, independence, and probability distributions, and also a mathematical language for communicating about these
ideas in the AP Statistics course. An understanding of sampling distributions as probability distributions
and statistics as random variables, together with a basic knowledge of the algebra of random variables,
provides a conceptually solid foundation for future study in college. Students should have a working
vocabulary that allows active classroom discussion of statistical ideas based on the grounding of probabilistic ideas.
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Scoring the Free-Response Questions
4. Statistical Inference Occurs in a Very Real World
Successful students should have a firm grasp of the nature and logic of statistical inference as it unfolds in
scientific studies, from planning to p-value. Random sampling provides the basis for generalizing one’s
findings beyond the data at hand. Proper experimental design is a tool for controlling extraneous variables
and reducing the ambiguities of experimental results. The formal inferential techniques of confidence
intervals and hypothesis testing lead to the assessment of the statistical significance of particular experiments.
5. Communication of Analysis, Methods, and Results
Even the best experimental design and statistical analysis will suffer if the reporting of methods and results
is incomplete, ambiguous, or misleading; student responses on the AP Statistics open-ended questions are
not exempt from this reality. It may seem paradoxical to demand precision and clarity in a science devoted
to the study of uncertainty, but the communication demands in modern statistical practice are congruent
with the test situation. Statistics, more than any other mathematical discipline, features communication
between expert and relative novice. The practice of modern statistics involves explanation, interpretation,
and translation. Students should be prepared to justify their analyses to a statistically literate reader as well
as write cogent explanations for general public consumption.
The Communication Dimension of the Free-Response Questions
The arrival of the National Council of Teachers of Mathematics’ (NCTM) Principles and Standards has
focused the attention of mathematics teachers on problem solving and writing in the math classroom.
Mathematics teachers are very familiar with exhorting students to “show their work” on their homework
and exams, and, of course, writing is a standard method of communication in school. However, this
new focus brings math teachers and math students to the frontier of their mathematical knowledge
and their ability to communicate fluently in writing. There are few guidelines and no ironclad rules on
this boundary.
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Scoring the Free-Response Questions
Statistics is a discipline where communication at the boundary is an essential skill. Consulting statisticians as well as investigators in the various scientific fields must be able to formulate real research questions
into a statistical form and then interpret the results of their statistical analysis for others. The evaluation of
student responses on the free-response section of the AP Statistics Exam reflects the importance of communication.
The AP Central Web site contains free-response questions and scoring rubrics from past AP Statistics
Exams. This is an excellent place to become more familiar with specific implementations of the general
rubric template for specific free-response exam questions. However, the specificity of the rubrics on past
questions can be misinterpreted. The level of detail of the rubrics and their association with the specific
question at hand renders them not completely reliable as general guidelines. After reading the individual
rubrics, one might be tempted to create overly-specific rules for students’ statistical writing. For example, a
particular hypothesis-testing question may ask students to comment on how the data were collected, and
this would, of course, be reflected in the rubric. A cursory reading of the rubric might lead one to think
that for all hypothesis-testing questions, a discussion of the sampling is required. This temptation must be
resisted! A particular element in the response to, say, an inference question could be crucial, less than
crucial, or even irrelevant in a different question.
While it may never be possible to completely codify guidelines or rules of thumb for approaching the
free-response questions, it is now possible to offer some general observations about common limitations
and errors of omission and commission in student responses on past exams. Writing in statistics is a matter
of judgment, wisdom, and experience, and these qualities tend to be in short supply while one is learning
the basics. Judgment and wisdom will arrive in due time; we can provide some benefits of experience now.
1. General Remarks
A common omission by students is the failure to define symbols correctly. With the time constraints of an
exam, students may forget which Greek letter is the appropriate one to use and provide, without explanation, an incorrect symbol. Defining the meaning of a symbol will, in general, inoculate students against
ambiguity or the presumption of error. For example, defining dA to be the population mean effect of drug
A is not the standard way of proceeding, but it does eliminate ambiguity. Specifically, students should
distinguish between population parameters and their estimates verbally or with symbols. They should
mention “the population mean” or “the sample mean,” rather than simply “the mean.”
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Scoring the Free-Response Questions
As a general rule, students should communicate in the accepted symbolism and format of statistical
writing when writing their responses. Frequently, students have attempted to communicate using the
symbolism and format of their calculators. For example, some calculators will perform a hypothesis test
automatically and present the results on the screen in a format that is different from the accepted statistical
format. Students may then simply copy what they see on their calculator screens. This is seldom an appropriate form of communicating a process and should be avoided.
Sometimes students will write too much or too little. The best strategy is to (a) clearly answer the
question asked and (b) stop. It frequently seems that students begin their writing while still formulating
their thoughts, perhaps under the impression that they are using their time more efficiently. Also, some
students seem to feel they must fill up the space allotted for the question. These two strategies have a
common effect: irrelevant or possibly incorrect or contradictory communication at either the beginning of
the response or the end. A much better strategy is to read the question carefully and then respond to that
question. Contradictory writing will always be regarded as incorrect. Should two parallel solutions be
offered, both will be read, but the lesser/least score will be awarded.
A common omission is the failure to interpret the results of a statistical procedure. Most questions on
the AP Statistics Exam require students to use a statistical procedure in a context. Student responses are
evaluated on the statistical methodology and the interpretation of the results. Students may feel that the
statistical methodology will carry the day and the reader of the response will “know what was meant.”
Irrespective of what was meant, every response will be evaluated on what was written. If a question is asked
in a context, students must explicitly interpret the statistical results in that context.
2. Exploring Data
The graphic presentation of data is, of course, an excellent method of communicating about distributions
of data and relations between variables. Effective communication with graphs will be compromised without correctly labeled and scaled axes. In comparative displays, students frequently fail to label which
group is associated with which display. Another common error is the use of different scales in a comparative setting. For example, the graphic clarity of comparative boxplots of the heights of males and females
is destroyed if each boxplot has its own scale. Descriptions of distributions should, as a matter of course,
address the center, variability, and shape as well as unusual features like outliers, gaps, and clusters.
When interpreting such distributions, these features should be explicitly mentioned; when comparing
distributions, characteristics of each distribution should be explicitly mentioned. To say that distribution
A is skewed to the right, for example, does not carry any implicit information that distribution B is not
so skewed.
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Scoring the Free-Response Questions
3. Planning Studies
Writing responses to questions about planning studies has historically been difficult for students. This is
not surprising since these questions tend to elicit the least formulaic responses. A student who, for example, struggles with hypothesis testing can demonstrate some knowledge by adhering to the stylistic form
of the hypothesis testing procedure. The “writing style” for discussion of methods of data collection and
experimental design is almost blank verse by comparison. Some students seize this opportunity and write
what are, in effect, blank responses, though they may entirely fill the allotted space for the response.
Possibly the best advice for responding to these questions is to know the vocabulary of sampling and
experimental design and to write with precision using that vocabulary. With questions about experimental
design in particular, it is not uncommon for students to answer questions that were not asked, leaving less
time and space to respond effectively to the question that was asked. For example, if asked to identify a
potential confounding variable in a particular scenario, it is not needed, nor is it necessarily helpful, for the
student to define what a confounding variable is. Some students adopt a shotgun strategy and attempt to
regurgitate what they remember about confounding variables in the hope that in the process of writing
their response they will stumble upon something relevant to the question. It is more likely, with holistic
grading, that the lack of focus in their answer will count against them.
4. Probability
Probability questions are the most “math-like” on the AP Statistics Exam, and students who know how to
solve these problems should be relatively comfortable doing so. Difficulties in this area tend to be more
organizational than conceptual, and students can improve their chances of doing well by using some
simple strategies.
a) Establish the formula first. Students should state what formula they are using in their calculations and substitute values from the problem appropriately. At that point, not before, students
should pick up their calculators.
b) Work through the entire problem. If a problem requires a sequence of calculations, intermediate values should be shown. Generally, if a question has multiple parts and a student misses an
early part, the evaluation of successive parts will presume that whatever answer was given earlier is
correct. That is, an error in part (a) will not automatically invalidate answers in parts (b), (c), and
so on. Giving credit in the face of earlier errors depends on being able to trace the thread of
reasoning through the problem.
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Scoring the Free-Response Questions
c) Use graphics. Students should take advantage of graphic representations of the problem in their
response. Displays of Venn diagrams and tree diagrams can effectively convey the students’
understanding of probability problems as well as reduce ambiguity in the algebraic presentation of
their responses.
d) Answer in context. Because probability questions are math-like, it seems to be easy for students
to forget that the question is presented in a context. They should not stop at getting a numerical
answer; they should ensure the numerical answer is embedded in the context of the problem in a
complete sentence.
5. Inference
Answers to inference questions tend to be stylistic in nature, although presentations in textbooks are
slightly different. Complete (i.e., score = 4) responses to inference questions where students use hypothesis
testing will generally require five components.
a) The correct statement of a set of hypotheses. Null and alternative hypotheses must be stated,
correctly defining any notation used. The use of H0 : , w and are standard symbols and need
not be defined. However, the use of the symbols should be specified (e.g., “Let Q A represent the
population mean height of corn under treatment A.”). The symbols as presented on the formula
sheet should be regarded as defined and meaningful, and other notation should be defined before
use. A common error is to specify the symbols, Qa , Q b , but then fail to link them to the particular
populations under discussion. This is particularly unfortunate in the case of a one-tailed hypothesis test.
b) Identification of an appropriate test procedure. The procedure and test statistic must be
clearly specified. This should be in the usual statistical form, avoiding the “calculator form”
naming the procedures for button pushing. The correct formula for the test statistic is usually the
least ambiguous identification of a procedure, although a construction such as “Two-sample t-test
for independent samples” is also acceptable.
c) Identifying and checking the appropriate assumptions. An important part of using inferential
procedures is recognizing the assumptions that underlie and justify the procedures. Different texts
will counsel slightly different requirements, and no specific requirement is enforced by the question rubrics except that it be commonly accepted in the statistical community. For example,
requirements for using the large-sample test for a single population proportion are usually stated
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Scoring the Free-Response Questions
in the format np u k , n(1 p ) u k , but the actual value for k will differ. It is not important that
students identify a “correct” value for k. What is important is that they know there are assumptions that underpin inferential procedures, recognize which are appropriate for the given problem,
and check them.
A common omission students make is the actual checking of the assumption by substituting
values for n, p, k, and stating that the conditions are met. Simply stating the assumptions without
checking them explicitly is not sufficient for a complete response. A top-notch approach to
checking assumptions is to include a statement about the reason for checking. For example, a
common check for the credibility of a normal population is graphic: constructing a boxplot. If
there is no indication of skewness, the t-procedure may be judged as justified. A student who
sketches the boxplot, comments that there are no outliers or other indications of skew, and then
links those comments to the appropriateness of the t-procedure, would have a stellar check of the
assumptions.
Many students show some confusion about the assumption of normality. This assumption
refers to the population from which the data is drawn. Specifically, the assumption is not that
there are no outliers in a set of data, it is that the population from which the data is drawn may
credibly be presumed to be normal, or at least approximately so. In addition, the evidence must
be linked to the assumption. Simply stating, for example, that there are “no outliers” is not
sufficient for establishing normality.
It is often forgotten that random sampling from a known population is an assumption for
making generalizations to that population. In addition, random assignment to treatments is
required for an argument of causation based on differences in experimental groups. Students
should be sensitive to these concerns when responding to inference questions in contexts and
express reservations when appropriate.
d) The correct mechanics. This is usually a matter of showing intermediate steps in an organized
manner. The correct values of the test statistic, the number of degrees of freedom when appropriate, the level of significance, and the p-values appropriately identified in the body of the work
is essential.
e) Stating correct conclusions in the context of the problem. Test statistics must be used to arrive
at a correct conclusion, either via the p-value approach or a rejection region approach. Linkage
between the conclusion and test result should be stated clearly (e.g., “Because the p-value is less
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Scoring the Free-Response Questions
than 0.05, the null hypothesis is rejected.”). Of critical importance is the conclusion stated in
the context of the problem, consistent with the defined hypotheses. That is, a rejection of a null
hypothesis should be correctly interpreted in context.
If confidence intervals are used for making inferences about population parameters, the considerations
above will still apply. One particularly common omission that occurs when confidence intervals are used is
the checking of assumptions for the procedure. Both confidence intervals and hypothesis testing are based
on the characteristics of the sampling distributions of the statistics of interest, and both procedures require
checks of the assumptions. If confidence intervals are used, a correct confidence interval statement is
required. Students frequently err by making a probability statement or indicating they have confidence
that a sample value rather than a population value is in a certain interval.
Rounding Answers
Remind your students that it is best not to round numbers at intermediate steps in a calculation. Answers
should be rounded only at the end and then not too much. For example, a z-score should be rounded to
two decimal places in order to use the table, but each step in the calculation should not be rounded at all,
or the final z-score may be quite different from what it would have been otherwise. A rule that is usually
satisfactory is to round the final answer to one more decimal place than was given in the data.
Defining Symbols
Students should define all symbols when writing solutions to open-ended questions. For example, when
writing a null hypothesis, students should not write just
Q ! 75,
but should state what Q represents. A clear and complete statement of a null hypothesis would be
Q ! 75, where Q is the mean of the reading test scores for all students in this school.
If students practice this policy throughout the course, they will have less trouble understanding that the
null hypothesis is about the population, not about the sample.
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Scoring the Free-Response Questions
Interpreting Sample Computer Output
The Development Committee strongly recommends the use of computers in performing data analysis. A
variety of inexpensive statistical packages exist, and the output from these programs is, for the most part,
standardized, though some slight variations in terminology may be seen. There may also be some differences in the information presented and its organization on the computer screen or printout. Since computers are not allowed in the actual AP Statistics Exam and calculators typically present information in a
nonstandard manner, some questions on the exam contain computer output. The computer output varies
from question to question and is not from any particular statistical package. The computer output given
for a particular question may be “complete,” or it is possible that only partial output with more focused
information may be presented for the students’ consideration. Students must be able to interpret the
computer output in order to answer both multiple-choice and free-response questions. Some examples of
output from some standard packages are shown in this section. As you will notice, the different software
packages present their output slightly differently and sometimes give some different information, so you
should ensure that your students have seen several different types of printouts before taking the exam. Such
sample output can be found in many introductory textbooks.
The data in this example come from a study of alarm barks in prairie dog towns (Motiff 1980).
These creatures emit repetitious alarm barks when there are intruders or other dangers present. The regression output presented is from fitting the bark frequency (barks/30 sec) vs. intruder distance from the
burrow (feet).
Bark Freq. of Black-Tailed
Prairie Dogs
Distance
Bark
from Burrow
Frequency
10.00
81
20.00
79
30.00
78
40.00
73
50.00
72
60.00
71
70.00
59
80.00
71
90.00
67
100.00
64
110.00
57
120.00
55
130.00
41
Bivariate Fit of Bark Freq. by Distance
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Scoring the Free-Response Questions
Regression Analysis from Minitab
Regression Analysis
The regression equation is
Bark Freq = 85.3 – 0.264 Distance
Predictor
Constant
Distance
S = 5.001
Coef
85.269
–0.26429
StDev
2.942
0.03707
R-Sq = 82.2%
T
28.98
–7.13
P
0.000
0.000
R-Sq(adj) = 80.6%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
1
11
12
SS
1271.2
275.1
1546.3
MS
1271.2
25.0
F
50.83
P
0.000
Unusual Observations
Obs
13
Distance
130
Bark Fre
41.00
Fit
50.91
StDev Fit
2.62
R denotes an observation with a large standardized residual.
74
Residual
–9.91
St Resid
–2.33R
Scoring the Free-Response Questions
Regression Analysis from Data Desk
Dependent variable is: Bark Frequency
R squared = 82.2%
R squared (adjusted) = 80.6%
s = 5.001 with 13 - 2 = 11 degrees of freedom
Source
Regression
Residual
Sum of Square
1271.21
275.093
Variable
Constant
Distant
Coefficient
85.2692
–0.264286
df
1
11
s.e. of Coeff
2.942
0.0371
Mean Square
1271.21
25.0085
t-ratio
29.0
–7.13
F-ratio
50.8
Prob
0.0001
0.0001
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Scoring the Free-Response Questions
Regression Analysis from JMP-Intro
Linear Fit
Bark Freq = 85.269231 - 0.2642857 Distance
Summary of Fit
Rsquare
Rsquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.822097
0.805924
5.000849
66.76923
13
Analysis of Variance
Source
Model
Error
C Total
DF
1
11
12
Sum of Squares
1271.2143
275.0934
1546.3077
Mean Square
1271.21
25.01
F Ratio
50.8313
Prob > F
<0.0001
Parameter Estimates
Term
Intercept
Distance
76
Estimate
85.269231
–0.264286
Std Error
2.942242
0.037069
T Ratio
28.98
–7.13
Prob > |t|
<0.0001
<0.0001