7 min read•july 11, 2024

Jerry Kosoff

FRQ practice is one of the most important ways to prepare for the exam! Sudent writing practice samples and corresponding feedback from Fiveable teacher Jerry Kosoff.

A large university offers undergraduate courses both in-person on its campus and digitally through an online platform. The administration of the university wishes to gauge their students’ opinions about the quality of teaching at the university; specifically, they want to estimate the proportion of all students who would agree that they are receiving good instruction. They are considering several methods for collecting this data.

- One method they are considering is conducting a simple random sample. The university has 30,000 students, and the administrators wish to ask a sample of 500 students for their opinion of the quality of teaching. Describe a procedure for how the administrators could conduct a simple random sample in this situation.
- A faculty member of the statistics department suggests to the administrators that they may want to consider a stratified random sample, using the primary type of instruction a student receives (in-person or digitally) as a variable for stratification. Describe in the context of the scenario why a stratified random sample may provide a more precise estimate of the proportion of all students who would agree that they are receiving good instruction than the simple random sampling method you detailed in #1.

Free Response Question Sample Submission 1

- Write the names of all 30,000 students on equal sized pieces of paper and put in a hat. Shake the hat. Then choose 500 pieces of paper out of the hat without replacement and collect the data they need.
- In a stratified random sample, you are guaranteed to get results from students who receive instruction online and in person. In a random sample, you have the ability to possibly get students from only in school or only online instruction. With a stratified random sample, you’d sort the students based on how they receive their instruction content. Then write down the names of each student in each strata on equal sized pieces of paper and place in a hat. Shake the hat. Then choose 250 students from the digital hat and 250 students from the in person hat and study their results. This will ensure that both populations of interest are being examined.

FRQ teacher feedback:

For #1, your use of the “names in a hat” method of conducting an SRS is clear, but stumbles a bit at the end: “collect the data they need” is a little vague and doesn’t connect to this scenario. It should be clear that the 500 pieces of paper representstudents(you do this in your first sentence) and that the selected students will be given the survey about quality of instruction. That second part isn’t clear in your response. On some rubrics, your response would still earn “E” (full credit); on others, it may be in jeopardy of being bumped to a “P” (partial credit)

For #2, you do a good job of explaining one purpose of a stratified sample (you are guaranteed to get results from each type of student), how that differs from an SRS (you might only get one group or the other), but then you describe how toimplementthe stratified sample. That wasn’t what the question asked: they want to knowwhythe stratified sample is a good idea. The beginning of your answer starts down that road, but should go a little further in describing the fact that having results from the two different groups of students is goodbecause their opinions about the quality of instruction may differ, which is the intent of the survey. Therefore, your response would likely get partial credit (“P”)

Free Response Question Sample Submission 2

- The administrators would randomly number the 30,000 students an integer between 1 and 30,000. Then, the students with the numbers 1-500 would be the sample of 500 students who would be asked about the opinion of the quality of teaching.
- Since students’ opinions of their quality of teaching might be similar within their primary type of instruction, the primary type of instruction a student receives (in-person or digitally) should be used as a variable for stratification. Stratification by the primary type of instruction should result in more precise estimates of student opinion than a simple random sample of the same size as there will be much less variability.

FRQ teacher feedback:

For part #1, I’m a little torn on whether you would earn “E” or “P”. Typically, the criteria for describing implementing a process like this are (1) clearly assign numbers to each individual, (2) generate a list ofnunique numbers within the boundaries of the assigned numbers, (3) select individuals who correspond with the numbers. It is clear that you have fulfilled components (2) and (3)… what I’m stuck on is (1). Saying “randomly number the 30,000 students [with] an integer between 1 and 30,000” does not clearly indicate that each student is receiving a unique number label; your description being just “randomly” leaves open the possibility of multiple people being assigned the same number, for example. This can be alleviated by using a clear randomization method. For example, you could have said “From a list of the 30,000 students, randomize the order of names and then assign the first name on the list 1, second name 2, and so on until all students are numbered. [rest of your response here]”. Or - and here’s the annoying part - simply add the word “unique” before “integer between 1 and 30,000”, and you’re covered. So with all of that said (sorry for long-winded response), you’d likely earn “P” for this part.

Much shorter feedback for #2: you crushed it. You show a clear understanding ofwhystratification is useful in this context, and gave a reason why (“opinions… might be similar within their primary type of instruction”). “E” for this part!

Free Response Question Sample Submission 3

- The administrators could conduct a simple random sample in this situation by randomly assigning the students a number from 1-30,000 (only 1 number per student). They can use a random number generator to select 500 students without replacement who can be asked for their opinion of the quality of teaching at the university.
- A stratified random sample may provide a more precise estimate of the proportion of all students who would agree that they are receiving good instruction than the SRS method in part (a). Because the variable for stratification is either if the student receives in-person or digital instruction, the spread of the data will be less (or less variability)

FRQ teacher feedback:

For #1, your response is strong, but missing one small component: when you use the RNG to select 500 students without replacement (as you should), you must specify that you want the RNG to select 500 numberswithin the range of 1-30,000. The way you’ve described it leaves us open to the possibility that 500 numbers will be generated, but not all 500 numbers will match the labels of students. Yes, it’s a small detail, but it’s been a part of rubrics for this type of question in the past. (The 3 things that are usually looked for: (1) give the population unique numbers, (2) generatenunique numbers within the range of the numbers from the population, (3) choose the [individuals] corresponding to those numbers and administer [thing].) Your response clearly does 2 of those 3 things, and would likely earn partial instead of full credit.

For #2, your response essentially summarizes the question (we’ll provide a more precise estimate of the proportion), without fully explainingwhythat happens. You mention that there will be “less variability”, but do not mention how stratifying the sample will do that. When explaining why we stratify (or block in experiments), it’s important to connect to the stratification/blocking variable to the response variable: in this case, that means explaining that the type of instruction a student receives may impact their opinion of the quality of that instruction (then insert a possible reason for this), which is why stratifying might help reduce variation: we’ll have separated the sample based on a factor that would influence the response, making our estimate more representative of the true population proportion.

Free Response Question Sample Submission 4

- The administrators should conduct a simple random sample by randomly assigning each student a unique number from 1-30,000. Then the administrators can use a random number generator and choose 500 numbers within the range of 1-30,000. The chosen 500 numbers would then represent the sample of 500 students who would be asked for their opinion of the quality of teaching by the administrators.
- The stratified random sample may provide a more precise estimate of the proportion of all students who would agree that they are receiving good instruction than the SRS method because students who receive in-person instruction may enjoy the instruction more that the people who receive online lectures. Therefore, the type of instruction received will influence a student’s response to the administrator’s question. Thus, the type of instruction received will be a confounding variable and result in variability in responses. To reduce variability, stratification of the type of instruction received will result in more precise estimates of students who agree they are receiving good instruction.

FRQ Teacher Feedback

Very well done! For #1, you check all of the boxes that we as readers must look for - each individual is given auniquenumber, the numbers you generate are within the bounds of the numbers you assign, and you clearly indicate what is being done with the individuals selected.

For #2, you give a clear description of why stratification is done, in the context of the scenario. Nicely done!

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