Ap Stats Review - Experimental Design

AP STATS REVIEW - EXPERIMENTAL DESIGN Multiple Choice

1) D 2) C 3) E 4) D 5) E 6) D 7) D 8) C 9) C 10) B 11) B 12) B 13) B (2013P #3) Part (a): The variable of interest is the number of bottles/cans per one-mile segment. The parameter of interest is the mean number of bottles/cans per one-mile segment of all roads in the state. Part (b): This survey was conducted as a stratified random sample with three strata corresponding to the three road types: major highways, minor highways, and unpaved roads. Separate random samples of 30 one-mile segments were taken within each of the three strata. Part (c): The Method I estimate is an unbiased estimate of the mean number of discarded cans and bottles per mile of public road in the state. The Method I estimate weights the sample means according to the proportion of each type of road in the population. The Method II estimate is biased in under-estimating the population mean. By giving equal weight to each type of road, the Method II estimate assigns too little weight to the minor highways and the minor highways appear to have more discarded cans and bottles than the other two types of roads.

Part (a) is scored as follows: Essentially correct (E) if both components (identifying the variable of interest and population parameter) are correct. Partially correct (P) if only one component is correct. Notes: 1. The definition of the parameter must include wording that would distinguish it from a sample mean. 2. The response does not need to include “per mile” in the definition of the variable or parameter

Part (b) is scored as follows: Essentially correct (E) if the student identifies a stratified random sample and provides a reasonable explanation that includes identification of road types as strata and indicates that a sample is selectedfrom each stratum. Partially correct (P) if the student identifies a stratified random sample but provides an incomplete explanation (such as stating that the state highways were grouped into three types but failing to clearly indicate that separate random samples of one mile road segments were taken within each group). OR The response clearly shows that the student knows the difference between cluster and stratified sampling by discussing both types, but has the names of the two types of sampling reversed. Incorrect (I) if the student identifies a stratified random sample with no justification or an incorrect justification OR chooses an incorrect sampling method. Part (c) is scored as follows: Essentially correct (E) if student chooses Method I and provides a justification that explicitly indicates that the strata sizes are different (for example, the state has more miles of minor highways than miles of major highways and unpaved roads) and therefore weighting by strata size is preferable to the unweighted average. Partially correct (P) if a student chooses Method I but provides a weak justification (one that does not explicitly indicate that the strata sizes are different, such as “it adjusts for road types” or “it uses the mileage proportions”). Incorrect (I) if the student chooses Method 1 with no justification or an incorrect justification OR chooses Method II. (2001 #3) Part (a): 1. Scheme: Obtain a two-digit random number from the random number table. If it is between 01 and 50, use it to represent the selected ticket. Ignore numbers 00 and 51 – 99. 2. Stopping Rule: Determine the amount of the prize associated with the chosen ticket, and add this amount to the total amount awarded so far. If the total amount awarded so far is less than $300, repeat this process. 3. Count: Note the total number of winners. 4. Non-Replacement: Ignore any ticket number that has already been awarded a prize in this trial. Repeats steps 1 – 4 above a large number of times. Note: It is OK to also devise a scheme that uses 2 two-digit numbers to represent each ticket (for example, 01 and 51 both representing ticket 1; 02 and 52 both representing ticket 2; etc.) that also addresses the issue of assigning 2 two-digit numbers to each coupon correctly. Part (b): Solution will depend on answer to part (a). For example, using scheme above:

There are five components to the solution of this problem: 1. Scheme: must include a clear, correct statement of the assignment of two-digit random numbers to the coupon numbers (and/or values) AND clear directions as to how the table is to be used in the simulation. 2. Stopping Rule: must state that the trial ends when a total value of $300 is attained or exceeded. 3. Count: must state or demonstrate that the number of winners is the outcome of the simulation. 4. Non-Replacement: must state that coupon numbers chosen cannot be used more than once in the same trial. 5. Execution of #1 and #2: must demonstrate a correct execution of a scheme with a stopping rule. Credit for components #1 and #2, is given for statements in part (a). Credit for component #3 and/or #4 may be given for statements or demonstrations in parts (a) or (b). Credit for component #5 is given for a clear demonstration in part (b). (2006B #5) Part (a): The response variable was the amount of draft. The two treatments were the standard hitch and the new hitch. The experimental units were the two large plots of land. Part (b): Yes, the two hitches (treatments) were randomly assigned to the two plots (experimental units). Part (c): No, each treatment (type of hitch) was applied to only one experimental unit (plot of land). Replication is used to repeat the treatments on different experimental units so general patterns can be observed. There is no replication in this study. Part (d): Although 25 measurements were taken at different locations in the two plots, each hitch was used in one plot (experimental unit) only. Thus, if a difference in the draft is observed we will not know whether the difference is due to the hitch or the plot. In statistical language, the treatments (hitches) are confounded with the plots.

Part (a) is essentially correct (E) if the response variable, treatments, and experimental units are correctly identified.

Part (a) is partially correct (P) if two of the three components of the experiment are correctly identified.

Note: Responses to parts (b), (c) and (d) must be considered with respect to the experimental units identified in part (a).

Part (b) is essentially correct (E) if the student correctly discusses the use of randomization in this experiment with respect to assignment of the two hitches for use in the two plots OR with respect to the experimental units identified in part (a).

Part (b) is partially correct (P) if the student recognizes the use of randomization in the experiment but provides an incomplete or unclear discussion.

Part (c) is essentially correct (E) if the student recognizes that replication was not used properly and provides a correct justification.

Part (c) is partially correct (P) if the student recognizes that replication was not used properly but provides an incomplete justification that reveals some understanding of replication.

Part (d) is essentially correct (E) if the student provides a valid explanation of confounding in this experiment.

Part (d) is partially correct (P) if the student provides an incomplete explanation that indicates an understanding of confounding in this experiment. For example, the student indicates that differences in plot conditions can affect draft but fails to link this to the inability to distinguish between plots differences and hitch effects.

Alternative solutions for part (d): Each treatment was used in only one plot. Therefore, any differences caused by the differences in plots (e.g., soil hardness, moisture level, etc.) cannot be separated from differences in the two treatments. Because only one plot of land is assigned to each hitch, if no difference is found it could be due to a superior hitch in a poor field (highly compacted) being compared with an inferior hitch in a good field. The effect of the hitch is masked by the differences in the plot (2009 #3)

Part (a) (completely randomized design): Each student will be assigned a unique random number using a random number generator on a calculator, statistical software, or a random number table. The assigned numbers will be listed in ascending order. The students with the lowest 12 numbers in the ordered list will receive the instructional program that requires physically dissecting frogs. The students with the highest 12 numbers will receive the instructional program that uses computer software to simulate the dissection of a frog. Part (a) alternative (randomized block design): Students will be paired or placed into blocks of size two, based on having similar pretest scores. So, the first block will contain the two students with the two lowest pretest scores, the second block will contain the two students with the third- and fourth-lowest pretest scores, and so on, with the last block containing the two students with the two highest pretest scores. In each block, the students will be assigned a unique random number using a random number generator on a calculator, statistical software, or a random number table. The student in each block with the lower random number will receive the instructional program that requires physically dissecting frogs, and the student with the higher random number will receive the instructional program that uses computer software to simulate the dissection of a frog. Part (b): By not randomizing and allowing the students to self-select, there is a potential for changes to occur in the differences between pretest and posttest scores for a particular group because of the characteristics of students who choose a particular instructional method, not because of the instructional method itself. For example, suppose frog-loving students already know a lot about frog anatomy; one would therefore expect these students to be less likely to show a large change between the pretest and posttest scores. Suppose the frog-loving students tend to select the computer simulation method (perhaps because they do not like the notion of dissecting the frogs they love). The possible low change between pretest and posttest scores for the computer simulation group might then be attributed to the students’ already knowing a lot about frog anatomy beforehand, not to the instructional method itself. The frog dissection group might see a larger change in scores because the students entering this group are those with the lower pretest scores (less prior knowledge) and who are thus more likely to show greater improvement between pretest and posttest scores.

Part (a) is scored as follows:

Essentially correct (E) if a proper method of randomization is described that (1) creates two groups of equal size; AND (2) assigns the named treatments to the groups in a manner that knowledgeable statistics users would employ to assign the students to the two instructional groups.

Partially correct (P) if only one of the two criteria above is met.

Notes:  Coin tossing (or equivalent method) using a stopping rule to obtain equal sample sizes requires placing the students in the class in a random order. If this method does not include a random order, at best, part (a) is scored as partially correct.  In using a random number table, if numbers are specified, the student must work with two-digit numbers. For example, if using the first 24 integers, the student must use 01–24, not 1–24. If the student uses numbers such as 1–24, a solution that would otherwise be essentially correct becomes partially correct, and a partially correct response becomes incorrect.

Part (a) alternative is scored as follows:

Essentially correct (E) if (1) blocks are formed based on students’ having similar pretest scores; AND (2) the two students in each block are assigned to different treatments; AND (3) the method of randomization used to assign the students in each block to the treatments is correct and can be implemented after reading the student’s response (in a manner that knowledgeable statistics users would employ to assign the students to the two instructional groups).

Partially correct (P) if two of the three components above are presented correctly.

Part (b) is scored as follows:

Essentially correct (E) if (1) the example gives a reasonable characteristic of the self-selected students in the study; AND (2) explains how this characteristic could be associated with changes in the differences between the pretest and posttest scores. Partially correct (P) if (1) the example gives a reasonable characteristic of the self-selected students in the study; AND (2) a weak explanation is provided of how this characteristic could be associated with changes in the differences between pretest and posttest scores.

Note: A weak explanation of how a characteristic could be associated with changes in the differences between pretest and posttest scores must at least mention test scores or state that one group will perform better than the other. (Simply mentioning a behavioral difference is not sufficient.)