Download AP Statistics Mixed Review Free Response

AP Statistics Mixed Review
Free Response
(2009B #5)
This problem is scored in four sections. Section 1 consists of part (a), step 1. Section 2 consists of part (a), steps 2
and 3. Section 3 consists of part (a), step 4. Section 4 consists of part (b). Sections 1, 2, and 3 are each scored as
essentially correct (E) or incorrect (I). Both step 2 and step 3 in section 2 must be completely correct to earn an E for
section 2. Section 4 is scored as essentially correct (E), partially correct (P), or incorrect (I).
Section 1 [part (a), step 1] is scored as follows:
Essentially correct (E) if the student states a correct pair of hypotheses.
Section 2 [part (a), steps 2 and 3] is scored as follows:
Essentially correct (E) if the student identifies a correct test (by name or formula) and includes correct mechanics.
Section 3 [part (a), step 4] is scored as follows:
Essentially correct (E) if the student states a correct conclusion in the context of the problem.
Section 4 [part (b)] is scored as follows:
Essentially correct (E) if the student finds the correct simulated p-value from the dotplot or states that the actual
standard deviation of 0.085 would be unusual if s = 0.05 because 0.085 lies in the tail of the distribution AND an
appropriate conclusion is made.
Partially correct (P) if only the simulated p-value or statement that 0.085 lies in the tail is correct, but
the conclusion is weak, wrong, or missing.
Notes:


If the p-value in section 2 is incorrect but the conclusion is consistent with the computed p-value, section 4
can be considered correct.
In section 4, if both an a and a p-value are given together, the linkage between the p-value and the
conclusion is implied. If no a is given, the solution must be explicit about the linkage by giving a correct
interpretation of the p-value or explaining how the conclusion follows from the p-value.
(2010B #5)
Part (a) is scored as follows:
Essentially correct (E) if the probability is computed correctly and appropriate work is shown OR the probability
calculation is set up correctly but a minor computational error is made.
Partially correct (P) if the probabilities of the two events are added without subtracting the probability of their
intersection, resulting in 1380/2500 = 0.552 .
OR
Independence is assumed in computing the probability of the intersection.
Note: An answer of 1135/2500 in fraction form is sufficient to be scored as essentially correct (E).
Part (b) is scored as follows:
Essentially correct (E) if the conditional probability is correctly computed and appropriate work is included OR if
the calculation is set up correctly but a minor computational error is made.
Partially correct (P) if the reverse conditional probability (being a college graduate given that he or she primarily
obtains news from the internet) is computed, resulting in 245/687 = 0.357 .
Notes
 An answer of 245/693 in fraction form is sufficient to be scored as essentially correct (E).
 A correct decimal answer with no work or justification is scored as incorrect (I).
Part (c) is scored as follows:
Essentially correct (E) if the response states that the events are not independent and gives a correct numerical
justification based on the table.
Partially correct (P) if the response states that the events are not independent and gives a correct statistical
justification, but numerical support is not included (for example, says that P(C/I) ≠ P(C) but never reports either
probability) OR the response includes correct and relevant calculations related to independence of these events but
reaches an incorrect conclusion that the events are independent.
Part (d) is scored as follows:
Essentially correct (E) if the chi-square test of association (or independence) is correctly identified and the correct
degrees of freedom are given.
Note: It is not necessary to show work in calculating the degrees of freedom.
Partially correct (P) if the response includes the correct name (chi-square test of association or independence) but
not the correct degrees of freedom.
Notes
 If the response includes only “chi-square test” without specifying “of association (or independence),” this
part is scored as essentially correct (E) provided that the degrees of freedom are computed correctly but as
incorrect (I) if the degrees of freedom are incorrect.
 If the response identifies the test as “chi-square test of goodness-of-fit” or “chi-square test of homogeneity
of proportions,” the response is scored as incorrect (I).
 If the response does not name a correct test and only gives correct degrees of freedom, the response is
scored as incorrect (I).
(2010 #1)
Part (a) is scored as follows:
Essentially correct (E) if the student correctly identifies all three subparts—the treatments, the experimental units
and the response that will be measured.
Partially correct (P) if the student identifies two subparts correctly.
Notes
 In subpart ii, it is acceptable to identify the experimental units as the cages.
 In subpart iii, it is not correct to identify the response as the mean number of granules consumed.
Part (b) is scored as follows:
Essentially correct (E) if the student produces a correct graph (a reasonable scatterplot or residual plot with correct
labels and scales) and then concludes, based on one or more features of the graph, that the pattern of the relationship
does not appear to be linear.
Partially correct (P) if the student produces a correctly shaped graph but concludes that the data are linear OR if the
student produces an incorrectly shaped graph but makes a reasonable conclusion based on one or more features of
the graph.
Note: Any of the following will result in an incorrect graph.
 Incorrect scale
 Reversed axes
 Missing label(s)
 Other types of graph (histogram, bar graph, etc.)
(2010 #4)
Part (a) is scored as follows:
Essentially correct (E) if the response correctly addresses the following two components:
 Calculation of the expected number of owners, showing a proper method for the calculation and providing
the correct numerical value
 Calculation of the standard deviation for the number of owners, indicating recognition of the appropriate
binomial distribution and providing the calculation and the correct numerical value
Partially correct (P) if the response contains only one of the two components listed above OR displays correct
formulas for both the expected value and the standard deviation of a binomial distribution but fails to show both of
the correct numerical values.
Part (b) is scored as follows:
Essentially correct (E) if the student does any of the following:
 Recognizes the applicability of the binomial distribution, identifies the correct parameters, sets up the
relevant probability calculation, and completes the calculation correctly
 Uses a normal probability approximation, identifying the relevant mean and standard deviation, and shows
a correct calculation of the probability
 Provides an argument based on an appropriate z-score, or the number of standard deviations away from the
mean, with a reasonable conclusion about likeliness
Partially correct (P) if the student does any of the following:
 Recognizes the applicability of the binomial distribution and identifies the correct parameters BUT sets up
an incorrect cumulative binomial probability calculation
 Recognizes the applicability of the binomial distribution and shows the calculation correctly BUT does not
identify the correct parameters in either part (a) or part (b)
 Recognizes the applicability of the normal approximation and identifies the correct parameters BUT
incorrectly calculates the z-score or probability
Notes
 If the parameter values were properly identified in part (a), they do not have to be identified in part (b).
 If the response shows a correct calculation of the probability, no comment about likeliness is necessary. But
such a comment is necessary if the response contains only a z-score without a probability or discusses
standard deviations from the mean.
 With the normal calculation, it is acceptable for the response to show the probability that the normal value
is below 11 or 11.5 or 12.
Part (c) is scored as follows:
Essentially correct (E) if the response describes an appropriate sampling method (e.g., stratified random sampling)
that ensures all of the following:
 Total sample size of 2,000
 At least 12 owners for each of the five car models
 Random selection of owners
Partially correct (P) if the response mentions stratified random sampling but gives a weak description, or no
description, of how to implement the procedure OR describes another appropriate sampling method but includes
only two of the three components listed above.
(2011 #6)
Section 1 is scored as follows:
Essentially correct (E) if the response correctly includes the following three components:
1. Identifies the correct inference procedure.
2. Checks the randomness condition.
3. Checks the large sample size condition.
Partially correct (P) if the response correctly includes exactly two of the three components listed above.
Notes
 The identification of the procedure must include “z,” “proportion,” and “interval.”
 Stating the correct formula for a confidence interval for a proportion is sufficient for the first component.
 “Random sample given” is sufficient for the second component.
 To satisfy the third component, the response:
o Must check both the number of successes and the number of failures.
o Must use a reasonable criterion (for example, ≥ 5 or ≥ 10).
o Must provide numerical evidence
 Any statement of hypotheses, definitions of parameters, statements of populations, etc. should be
considered extraneous. However, if such statements are included and incorrect, this should be considered
poor communication in terms of holistic scoring.
 Any checks of reasonable conditions, such as independence of observations, sample size less than 10
percent of population size, 9,600 > 30, etc. should be considered extraneous. However, if a response
includes an incorrect condition, such as population normality, reduce the score in section 1 from E to P or
from P to I.
 Any reference to the central limit theorem should be treated as extraneous and not sufficient for the large
sample size condition.
Section 2 is scored as follows:
Essentially correct (E) if the response correctly includes the following two components:
1. Calculates the interval.
2. Interprets the interval, including a confidence statement and correct parameter, in context.
Notes
 The critical value for the confidence interval must be for 99 percent confidence.
 If the response includes an incorrect formula or has incorrect values substituted into the formula, then the
response does not earn credit for the calculation component, even if the final interval is correct.
 A response that makes minor arithmetic mistakes in the calculation of the interval is considered correct, as
long as the resulting interval is reasonable.
 A correct interval that is stated only in the interpretation is considered sufficient for the first component.
 To identify the parameter, the response must refer to the proportion “who would answer the question
correctly” or include a modifier for the proportion such as “population” or “true.” An interpretation about
the sample proportion (for example, “the proportion of students who answered correctly”) is not sufficient
for the second component.
 If the response provides only an interpretation of the confidence level instead of the confidence interval, the
second component is considered incorrect. If an interpretation of the confidence level is given along with
an interpretation of the confidence interval, both must be correct to be considered sufficient.
 A correct interpretation with an incorrect interval is sufficient for the second component.
Partially correct (P) if the response correctly includes exactly one of the two components.
Section 3 is scored as follows:
Essentially correct (E) if the response correctly includes the following two components:
1. In part (b) completes the tree diagram in terms of k.
2. In part (c) adds the correct results from the tree diagram.
Notes
 If a response states “not k” in the first box, the first component is considered incorrect.
 If the response to part (b) is incorrect, then part (c) is considered correct if the response is consistent with
the response from part (b) or if the response to part (c) is correct.


The response to part (c) does not need to show a simplified expression.
The response to part (c) can be expressed as a fraction with the sum of the four branches in the
denominator.
 A response to part (c) that adds the appropriate probabilities from the tree but has an error in the
simplification of the sum is still considered correct.
 If the response to part (c) is expressed as P(0.25 + 0.75k) or equivalent, the second component is
considered incorrect.
 If the tree diagram includes numbers only, adding the appropriate values is sufficient for the second
component, provided that the sum is between 0 and 1.
Partially correct (P) if the response correctly includes exactly one of the two components.
Section 4 is scored as follows:
Essentially correct (E) if the response correctly includes the following three components:
1. Equates the expression from part (c) to a numerical estimate from part (a).
2. Uses the endpoints from part (a) to calculate a reasonable interval.
3. Interprets the resulting interval, including a confidence statement and correct parameter, in context.
Partially correct (P) if the response correctly includes exactly two of the three components.
Notes
 Using the point estimate pˆ = 0.28 from part (a) or the endpoints of the interval (0.268, 0.292) from part (a)
is sufficient for the first component.
 A response that makes minor arithmetic mistakes in the calculation of the interval is considered correct, as
long as the resulting interval is reasonable.
 For the third component, the parameter must be the proportion of students who actually know the answer to
the history question.
 A response that creates a correct interval using linear transformations (of the point estimate and standard
error/margin of error) is equivalent to transforming the endpoints and therefore is sufficient for the first two
components.
(2011B #6)
Section 1 is scored as follows:
Essentially correct (E) if the response includes the following two components:
1. The response in part (a) is correct, as evidenced by the correct interpretation of the slope, in context.
2. The response in part (b) is correct, as evidenced by the identification of extrapolation as the reason that the
model should not be used AND the response is in context.
Partially correct (P) if only one of the two components listed above is correct.
Notes
 Part (a) is incorrect if the interpretation is not in context or if the interpretation does not acknowledge
uncertainty (for example, does not include “on average” or “about” or “approximately” or “predicted”
when referring to the increase in nitrogen removed).
 Ideally a correct solution would also include units and make it clear that it is the approximate increase for
each additional foot added to the buffer, but in the context of this larger investigative task, failure to do so
is not sufficient to make part (a) incorrect.
 Part (b) is incorrect if extrapolation is not identified as the reason or if the response is not in context.
Section 2 is scored as follows:
Essentially correct (E) if the response includes the following two components:
1. The response in part (c) states that the sampling distribution is normal AND provides a correct mean and
standard deviation.
2. The response in part (d) uses the correct mean and standard deviation of the sampling distribution — or
incorrect values carried over from part (c) — AND a correct critical value (1.96 or 2) to compute a correct
interval.
Partially correct (P) if only one of two components listed above is correct.
Notes
 Stating that the sampling distribution is approximately normal is acceptable for part (c).
 Part (c) is incorrect if the response does not state that the distribution is normal or if an incorrect mean or
standard deviation is given.
 Part (d) is incorrect if an incorrect critical value (for example a t critical value) is used or if an incorrect
mean or standard deviation — other than incorrect values from part (c) — is used in the computation of the
interval.
Section 3 is scored as follows:
Essentially correct (E) if study plan 1 is chosen in part (e), and the response demonstrates awareness of sampling
variation in the estimates of the slopes of the regression lines, AND this is clearly communicated in the context of
the two study plans.
Partially correct (P) if study plan 1 is chosen in part (e), and the response demonstrates awareness of sampling
variation in the estimates of the slopes of the regression lines, BUT the justification of the choice of study plan 1 is
not clearly communicated.
Section 4 is scored as follows:
Essentially correct (E) if the response specifies another study plan that uses eight buffer strips of at least three
different specified widths, and the response indicates in how many locations each width will be used, OR the
response makes it clear that at least three different buffer widths will be used and indicates that the buffer widths to
be used will be spread out over the range of interest.
Note: Specifying eight different widths is sufficient for an E in section 4.
Partially correct (P) if the response does not meet criteria for E, BUT the stated plan uses at least three different
widths. Widths need not be specified for a P.