2023 AP Statistics Free-Response Questions
Advanced PlacementFree ResponseMathematics
6 questions12 pts
6 gradable21 rubric criteria0 dependencies
q1
Question 1
Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. 1. As part of a study on the chemistry of Alaskan streams, researchers took water samples from many streams with temperatures colder than 8°C and from many streams with temperatures warmer than 8°C. For each sample, the researchers measured the dissolved oxygen concentration, in milligrams per liter (mg/l). The histogram shows the dissolved oxygen concentration (mg/l) in streams with temperatures colder than 8°C. The frequencies shown are: 2–3: 1; 4–5: 1; 5–6: 1; 6–7: 13; 7–8: 12; 8–9: 15; 9–10: 40; 10–11: 94; 11–12: 117; 12–13: 89; 13–14: 46. (a) The researchers constructed the histogram shown for the dissolved oxygen concentration in streams from the sample with water temperatures colder than 8°C. Based on the histogram, describe the distribution of dissolved oxygen concentration in streams with water temperatures colder than 8°C. For the dissolved oxygen concentration in streams from the sample with water temperatures warmer than 8°C, the researchers computed the following summary statistics: | Min | Q1 | Median | Q3 | Max | Mean | Std. Dev. | |---|---|---|---|---|---|---| | 2.10 | 4.39 | 5.43 | 6.12 | 13.45 | 5.54 | 1.64 | (b) The researchers computed the summary statistics shown in the table for the dissolved oxygen concentration in streams from the sample with water temperatures warmer than 8°C. Use the summary statistics to construct a box plot for the dissolved oxygen concentration in streams with water temperatures warmer than 8°C. Do not indicate outliers. (c) The researchers believe that streams with higher dissolved oxygen concentration are generally healthier for wildlife. Which streams are generally healthier for wildlife, those with water temperature colder than 8°C or those with water temperature warmer than 8°C? Using characteristics of the distribution of dissolved oxygen concentration for temperatures colder than 8°C and characteristics of the distribution of dissolved oxygen concentration for temperatures warmer than 8°C, justify your answer.
Rubric Criteria (3)
C1For part (a), the response describes dissolved oxygen concentration in Alaskan streams as a unimodal, left-skewed distribution with center between 11 mg/l and 12 mg/l, and includes at least one valid measure of spread (range 10–12 mg/l, IQR 1.5–2.5 mg/l, standard deviation 1.7–1.8 mg/l, or all observations between 2 mg/l and 14 mg/l) and an unusual feature such as potential outliers or a gap between 3 mg/l and 4 mg/l. To be essentially correct, context plus at least three of the four shape, center, spread, and unusual-feature components must be present; a partially correct response includes context plus two components, or at least three non-context components.
0.0000pts
C2For part (b), the response constructs a boxplot for dissolved oxygen concentration in warmer-than-8°C streams with the box beginning at Q1 between 4 and 4.5 mg/l, ending at Q3 between 6 and 6.5 mg/l, the median between 5 and 5.5 mg/l, and whiskers extending to the minimum between 2 and 2.5 mg/l and the maximum between 13 and 13.5 mg/l. To be essentially correct, four or five of the five boxplot components must be present; a partially correct response has exactly three components.
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C3For part (c), the response states that colder-than-8°C streams are healthier for wildlife and includes at least two of the following: a direct comparison of the centers, an indication that warmer streams are right-skewed while colder streams are left-skewed, and a direct comparison of spreads using corresponding approximate range, IQR, or standard-deviation values. To be essentially correct, the health statement and at least two comparison components must be present; a partially correct response includes the health statement and one comparison component.
0.0000pts
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q2
Question 2
2. A developer wants to know whether adding fibers to concrete used in paving driveways will reduce the severity of cracking, because any driveway with severe cracks will have to be repaired by the developer. The developer conducts a completely randomized experiment with 60 new homes that need driveways. Thirty of the driveways will be randomly assigned to receive concrete that contains fibers, and the other 30 driveways will receive concrete that does not contain fibers. After one year, the developer will record the severity of cracks in each driveway on a scale of 0 to 10, with 0 representing not cracked at all and 10 representing severely cracked. (a) Based on the information provided about the developer’s experiment, identify each of the following. • Experimental units • Treatments • Response variable (b) Describe an appropriate method the developer could use to randomly assign concrete that contains fibers and concrete that does not contain fibers to the 60 driveways. Suppose the developer finds that there is a statistically significant reduction in the mean severity of cracks in driveways using the concrete that contains fibers compared to the driveways using concrete that does not contain fibers. (c) In terms of the developer’s conclusion, what is the benefit of randomly assigning the driveways to either the concrete that contains fibers or the concrete that does not contain fibers?
Rubric Criteria (3)
C1For part (a), the response identifies the driveways as the experimental units, identifies concrete with fibers and concrete without fibers as the treatments, and identifies the rating of crack severity as the response variable. To be essentially correct, all three components must be present; a partially correct response includes exactly two.
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C2For part (b), the response describes a random process assigning the 60 labeled driveways to concrete with fibers or without fibers such that every possible assignment is equally likely, and the process assigns 30 driveways to each treatment. To be essentially correct, both randomization and equal-size-group requirements must be present; a partially correct response includes only one.
0.0000pts
C3For part (c), the response uses random assignment to conclude that the type of concrete caused the rating of crack severity and states the conclusion in the context of the concrete treatment or severity rating. A partially correct response gives the causal implication without context or states in context that random assignment limits confounding.
0.0000pts
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Question 3
3. Bath fizzies are mineral tablets that dissolve and create bubbles when added to bathwater. In order to increase sales, the Fizzy Bath Company has produced a new line of bath fizzies that have a cash prize in every bath fizzy. Let the random variable, X, represent the dollar value of the cash prize in a bath fizzy. The probability distribution of X is shown in the table. Cash prize, x: $1 | $5 | $10 | $20 | $50 | $100 Probability of cash prize, P(X = x): P(X = $1) | 0.2 | 0.05 | 0.05 | 0.01 | 0.01 (a) Based on the probability distribution of X, answer the following. Show your work. (i) Calculate the proportion of bath fizzies that contain $1. (ii) Calculate the proportion of bath fizzies that contain at least $10. (b) Based on the probability distribution of X, calculate the probability that a randomly selected bath fizzy contains $100, given that it contains at least $10. Show your work. (c) Based on the probability distribution of X, calculate and interpret the expected value of the distribution of the cash prize in the bath fizzies. Show your work. (d) The Fizzy Bath Company would like to sell the bath fizzies in France, where the currency is euros. Suppose the conversion rate for dollars to euros is 1 dollar = 0.89 euros. Using your expected value from part (c), calculate the expected value, in euros, of the distribution of the cash prize in the bath fizzies. Show your work.
Rubric Criteria (4)
C1For part (a), the response correctly calculates the proportion containing a $1 prize as 0.68 (or 68 percent) and shows supporting work, and correctly calculates the proportion containing at least a $10 prize as 0.12 (or 12 percent) and shows supporting work. The random variable is identified as the dollar value of the cash prize. The essentially-correct condition requires at least three of the four listed components; the partially-correct condition requires exactly two.
1.0000pts
C2For part (b), the response calculates the conditional probability that a bath fizzy contains $100 given that it contains at least $10 as 0.01/0.12, approximately 0.0833, with work consistent with the response for the proportion containing at least $10. A partially correct response satisfies only one of the correct-calculation or consistent-work requirements.
1.0000pts
C3For part (c), the response calculates the expected value as $4.68 and shows appropriate work using the probability-weighted prizes, including enough terms to show the pattern. The interpretation must include at least two of long-run repetition, average or mean, and cash-prize context for an essentially correct response; a partially correct response satisfies two or three of the listed components without meeting the essentially-correct condition.
1.0000pts
C4For part (d), the response converts the expected value to approximately 4.17 euros using the conversion factor 0.89, shows work, and includes units. A partially correct response has the correct euro value but omits either work or units.
1.0000pts
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q4
Question 4
4. A medical researcher completed a study comparing an omega-3 fatty acids supplement to a placebo in the treatment of irritability in patients with a certain medical condition. Nineteen patients with the medical condition volunteered to participate in the study. The study was conducted using the following weekly schedule. • Week 1: Each patient took a randomly assigned treatment, omega-3 supplement or placebo. • Week 2: The patients did not take either the omega-3 supplement or the placebo. This was necessary to reduce the possibility of any carryover effect from the assigned treatment taken during week 1. • Week 3: Each patient took the treatment, omega-3 supplement or placebo, that they did not take during week 1. At the end of week 1 and week 3, each patient’s irritability was given a score on a scale of 0 to 10, with 0 representing no irritability and 10 representing the highest level of irritability. For each patient, the two irritability scores and the difference in their scores (placebo minus omega-3) were recorded. The results are summarized in the table and boxplots. Table: Treatment | n | Mean | Standard Deviation Placebo | 19 | 5.421 | 2.987 Omega-3 | 19 | 3.632 | 1.739 Difference (placebo minus omega-3) | 19 | 1.789 | 2.485 Boxplots are shown for Placebo, Omega-3, and Difference in Irritability Score (Placebo Minus Omega-3), with the horizontal axis labeled Irritability Score or Difference in Irritability Score and marked from −3 through 10. The researcher claims the omega-3 supplement will decrease the mean irritability score of all patients with the medical condition similar to the volunteers who participated in the study. Is there convincing statistical evidence to support the researcher’s claim at a significance level of α = 0.05? Complete the appropriate inference procedure to support your answer.
Rubric Criteria (3)
C1For Section 1, the response identifies a paired or matched-pairs t-test for a population mean difference, states hypotheses in terms of a single mean, uses H0: μd = 0 (or μd ≤ 0) and Ha: μd > 0, and defines the parameter with the population mean difference in irritability scores (placebo minus omega-3) for people with the medical condition. A partially correct response satisfies three of these four components.
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C2For Section 2, the response checks independence by referring to random treatment assignment, supports approximate normality of the sampling distribution by indicating that the differences are not badly skewed or have no problematic outliers, reports a t-statistic consistent with the named test (approximately 3.138 for the paired approach), and reports a p-value consistent with the alternative and statistic (approximately 0.0028 for the paired approach). A partially correct response satisfies two or three components.
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C3For Section 3, the response compares the identified p-value with α = 0.05 and makes a consistent decision about the null or alternative hypothesis, then gives a non-deterministic conclusion in context stating evidence about the population mean irritability-score difference (placebo minus omega-3) for people with the medical condition. A partially correct response satisfies only the comparison-and-decision component or only the contextual conclusion component.
0.0000pts
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q5
Question 5
5. Wildlife biologists are interested in the health of tule elk, a species of deer found in California. An important measurement of tule elk health is their weight. The weight of a tule elk is difficult to measure in the wild. However, chest circumference, which is believed to be related to the weight of a tule elk, can easily be measured from a safe distance using a harmless laser. A study was done to investigate whether chest circumference, in centimeters (cm), could be used to accurately estimate the weight, in kilograms (kg), of male tule elk. For the study, wildlife biologists captured 30 male tule elk, measured their chest circumference and weight, and then released the elk. The data for the 30 male tule elk are shown in the scatterplot. [Scatterplot shown: Weight (kg) versus Chest Circumference (cm) for 30 male tule elk.] (a) Describe the relationship between chest circumference and weight of male tule elk in context. Following is the equation of the least-squares regression line relating chest circumference and weight for male tule elk. predicted weight = −350.3 + 3.7455(chest circumference) (b) The weight of one male tule elk with a chest circumference of 145.9 cm is 204.3 kg. (i) Using the equation of the least-squares regression line, calculate the predicted weight for this male tule elk. Show your work. (ii) Calculate the residual for this male tule elk. Show your work. The equation of the least-squares regression line relating chest circumference and weight for male tule elk is repeated here. predicted weight = −350.3 + 3.7455(chest circumference) (c) Interpret the slope of the least-squares regression line in context. (d) The sambar, another species of deer, is similar in size to the tule elk. The slope of the population regression line relating chest circumference and weight for all male sambars is 4.5 kilograms per centimeter. A wildlife biologist wants to determine whether the slope of the population regression line for male tule elk is different than that for male sambars. Let β represent the slope of the population regression line for male tule elk. The wildlife biologist conducted a test of the following hypotheses using the sample of 30 tule elk. H₀: β = 4.5 Hₐ: β ≠ 4.5 The test statistic was calculated to be 3.408. Assume all conditions for inference were met. (i) Determine the p-value of the test. (ii) At a significance level of α = 0.05, what conclusion should the wildlife biologist make regarding the slope of the population regression line for male tule elk? Justify your response.
Rubric Criteria (4)
C1For part (a), the response describes the association between chest circumference and weight of tule elk as positive or increasing, strong, and linear or approximately linear, indicates that no points seriously deviate from the straight-line pattern or that there are no unusual features, and provides context involving chest circumference and weight of tule elk. To be essentially correct, context plus at least three of the four direction, strength, form, and unusual-feature components must be present; a partially correct response has context plus one or two components, or at least two non-context components.
1.0000pts
C2For part (b), the response calculates the predicted weight for a male tule elk with chest circumference 145.9 cm as approximately 196.17 kg with work shown, and calculates the residual for actual weight 204.3 kg as approximately 8.13 kg, or a value consistent with the predicted weight used, with work shown. A partially correct response satisfies only one component or gives both correct values without showing work.
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C3For part (c), the response identifies the slope as 3.7455 (or an accepted rounding such as 3.746, 3.75, or 3.7), interprets it as an increase in predicted, estimated, expected, or average weight in kilograms for each additional centimeter of chest circumference, and correctly specifies units for weight and length. A partially correct response satisfies two of these three components.
1.0000pts
C4For part (d), the response reports a two-sided p-value of 0.002, compares it correctly with α = 0.05 and makes a consistent decision about the null or alternative hypothesis, and concludes in context, using non-deterministic language, that there is sufficient statistical evidence concerning whether the population regression slope of weight on chest circumference for male tule elk differs from 4.5 kg/cm. A partially correct response satisfies two of these three components.
1.0000pts
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q6
Question 6
Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. 6. A jewelry company uses a machine to apply a coating of gold on a certain style of necklace. The amount of gold applied to a necklace is approximately normally distributed. When the machine is working properly, the amount of gold applied to a necklace has a mean of 300 milligrams (mg) and standard deviation of 5 mg. (a) A necklace is randomly selected from the necklaces produced by the machine. Assuming that the machine is working properly, calculate the probability that the amount of gold applied to the necklace is between 296 mg and 304 mg. The jewelry company wants to make sure the machine is working properly. Each day, Cleo, a statistician at the jewelry company, will take a random sample of the necklaces produced that day. Each selected necklace will be melted down and the amount of the gold applied to that necklace will be determined. Because a necklace must be destroyed to determine the amount of gold that was applied, Cleo will use random samples of size n = 2 necklaces. Cleo starts by considering the mean amount of gold being applied to the necklaces. After Cleo takes a random sample of n = 2 necklaces, she computes the sample mean amount of gold applied to the two necklaces. (b) Suppose the machine is working properly with a population mean amount of gold being applied of 300 mg and a population standard deviation of 5 mg. (i) Calculate the probability that the sample mean amount of gold applied to a random sample of n = 2 necklaces will be greater than 303 mg. (ii) Suppose Cleo took a random sample of n = 2 necklaces that resulted in a sample mean amount of gold applied of 303 mg. Would that result indicate that the population mean amount of gold being applied by the machine is different from 300 mg? Justify your answer without performing an inference procedure. Now, Cleo will consider the variation in the amount of gold the machine applies to the necklaces. Because of the small sample size, n = 2, Cleo will use the sample range of the data for the two randomly selected necklaces, rather than the sample standard deviation. Cleo will investigate the behavior of the range for samples of size n = 2. She will simulate the sampling distribution of the range of the amount of gold applied to two randomly sampled necklaces. Cleo generates 100,000 random samples of size n = 2 independent values from a normal distribution with mean μ = 300 and standard deviation σ = 5. The range is calculated for the two observations in each sample. The simulated sampling distribution of the range is shown in Graph I. This process is repeated using σ = 8, as shown in Graph II, and again using σ = 12, as shown in Graph III. Graph I with σ = 5; Graph II with σ = 8; Graph III with σ = 12. Each graph displays Relative Frequency versus Sample Range. (c) Use the information in the graphs to complete the following. (i) Describe the sampling distribution of the sample range for random samples of size n = 2 from a normal distribution with standard deviation σ = 5, as shown in Graph I. (ii) Describe how the sampling distribution of the sample range for samples of size n = 2 changes as the value of the population standard deviation σ increases. Recall that Cleo needs to consider both the mean and standard deviation of the amount of gold applied to necklaces to determine whether the machine is working properly. Suppose that one month later, Cleo is again checking the machine to make sure it is working properly. Cleo takes a random sample of 2 necklaces and calculates the sample mean amount of gold applied as 303 mg and the sample range as 10 mg. (d) Recall that the machine is working properly if the amount of gold applied to the necklaces has a mean of 300 mg and standard deviation of 5 mg. (i) Consider Cleo’s range of 10 mg from the sample of size n = 2. If the machine is working properly with a standard deviation of 5 mg, is a sample range of 10 mg unusual? Justify your answer. (ii) Do Cleo’s sample mean of 303 mg and range of 10 mg indicate that the machine is not working properly? Explain your answer.
Rubric Criteria (4)
C1For part (a), the response uses a normal distribution with mean 300 mg and standard deviation 5 mg for the amount of gold applied to a randomly selected necklace, specifies the event that the amount is between 296 mg and 304 mg, and obtains probability 0.5763 or a value consistent with the stated distribution and event. A partially correct response satisfies two of these three components.
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C2For part (b), the response models the sample mean for two necklaces with an approximately normal distribution having mean 300 mg and standard deviation 5/√2 ≈ 3.5355 mg, specifies the event that the sample mean exceeds 303 mg, and obtains probability 0.198 or a value consistent with the stated distribution and event. It also states whether observing 303 mg provides convincing evidence that the population mean differs from 300 mg and justifies that decision using the probability from part (b-i) or a correctly computed corresponding probability. The essentially-correct condition requires components 1, 4, and 5 plus component 2 or 3; the partially-correct alternatives are those stated in the rubric.
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C3For part (c), the response states that the sampling distribution of the sample range for samples of size 2 from a normal population with standard deviation 5 mg is right-skewed, has center about 6 mg, and has most simulated ranges between 0 mg and about 25–30 mg. It also states that increasing the population standard deviation increases both the center or mean and the spread or variation of the sample-range distribution. An essentially correct response includes four or five of these five components; a partially correct response meets one of the combinations specified in the rubric.
1.0000pts
C4For part (d), the response states that a sample range of 10 mg is not unusual when the machine has population standard deviation 5 mg and justifies this using the sampling distribution, indicating that values of 10 mg or greater occur often. It states that Cleo's sample mean of 303 mg and range of 10 mg do not provide convincing evidence that the machine is not working properly, and justifies the conclusion using that the sample mean is not unusual (for example, probability about 0.198 or less than one standard deviation from 300 mg) and that the sample range is not unusual. An essentially correct response includes four or five of these five components; a partially correct response meets one of the combinations specified in the rubric.
1.0000pts
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This material is the intellectual property of College Board, is not owned by GradeThis, and is shown here for ease of use and demonstration purposes.