AP Stats Mixed Units Practice FRQ #3 & Feedback

📊AP Statistics Review

Practicing with FRQs is a great way to prep for the AP exam! Review student responses for a FRQ combining multiple units and corresponding feedback from Fiveable teacher Jerry Kosoff.

How This Connects to the AP Statistics Exam

On the AP Statistics Exam, the free-response section (Section II) has 6 questions total: 5 questions in Part A completed in 65 minutes, and 1 investigative task in Part B completed in 25 minutes. The full AP Statistics Exam lasts 3 hours total: Section I has 40 multiple-choice questions in 90 minutes, and Section II has 6 free-response questions in 90 minutes total, divided into Part A (Questions 1–5) completed in 65 minutes and Part B (Question 6, the investigative task) completed in 25 minutes.

Mixed-units FRQs often require you to identify an appropriate method, show required statistical work, check conditions when doing inference, and communicate conclusions in context. When practicing, write complete responses that match the task. For inference questions, this often means naming the procedure, defining parameters, stating hypotheses when doing a significance test, checking conditions, showing statistical work, and interpreting results in context.

This practice especially builds skills in data analysis, probability/sampling distributions, and statistical argumentation.

The Mixed Units FRQ Prompt

A group of randomly-selected students at a large high school were given a survey about their transportation to school. One of the questions asked students for their primary mode of transportation; another asked students for the typical number of minutes it takes them to travel to school in the morning. Two groups of students are singled out for further comparison: those that said they walk to school as their primary mode of transportation, and those that said they drive to school as their primary mode of transportation. The two histograms below show the distribution of travel times, in minutes, for the two groups - though the group labels are not present. There are a similar number of students in each group.

Summary statistics for the two groups revealed that for the students who walk to school, the mean time it typically takes to get to school is lower than the median time it typically takes to get to school. Based on this information, which of the histograms shown above represents the group of students who walk to school? Justify your response.
If the data from the two groups were combined into a single histogram, describe the shape of the resulting histogram.
The distribution of travel times for the 500 staff members at the school shows a moderate right skew, with a mean of 25 minutes and a standard deviation of 18 minutes. For random samples of 36 staff members, describe the sampling distribution of sample mean travel times.
A student wishes to determine if the mean travel time for staff members is different from the mean travel time for students at the school. To investigate, the student surveys 50 randomly-selected students and 40 randomly-selected staff members. State the name of the appropriate statistical test the student should conduct, and write appropriate hypotheses for the test you name. Be sure to define any parameters you use.

FRQ Writing Samples and Teacher Feedback

Student Response 1

Histogram K represents the group of students who walk to school because the mean is pulled towards outliers. Since Histogram K is skewed left, the mean would be pulled towards the skewness while the median would stay put as it is resistant to outliers. Histogram J however, shows the mean being greater than the median as it is being pulled to the right.
The resulting histogram of the combination of the two groups of students who walk to school and time it takes for school would be normally distributed.

Name of test: 2 sample t-test. We want to test a claim about the difference between mu1 (the true mean travel time it takes to travel to school for 50 students) and mu2 (the true mean travel time it takes to travel to school for 50 students.

Teacher Feedback

Your answer for part 1 is very good - you pick the correct histogram and give a clear reason for doing so (and clearly explain why it WOULDN’T be the other histogram). Full credit. For part 2, you say “normal,” which implies a unimodal graph with a peak in the center of the distribution. That won’t be the case here: if you look carefully at the x-axis for each distribution the peaks will end up separated (one around 8 minutes and the other around 40 minutes), whereas the center of the combined number line (around 24 minutes) has very few observations. Thus, bimodal is the description we’d be looking for here.

For part 3, your work is good - you’ve checked conditions, cited the use of the CLT, and correctly calculated the parameters of the sampling distribution. One thing - you never mention the shape (just saying “normality” as a condition isn’t equivalent to saying “the sampling distribution would be approximately normal.”). Therefore, you’d get partial instead of full credit. Advice - after citing the CLT, state that this means the sampling distribution is approximately normal.

Finally, in part 4, you’d also receive partial credit, if any. A 2-sample t-test is the correct test. However, you do not write fully correct statements for Ho and Ha. Your Ho would be that the mean travel times for the two groups are the same; Ha would be that the mean travel times for the groups are different. You only mention the ‘difference’ and do not cite it as the alternative hypothesis. Additionally, you define mu1 and mu2 as representing 50 students - but they would represent ALL students and ALL staff members at the school (since “mu” is a population parameter). Be careful when typing/writing responses!

Student Response 2

If the mean time is lower than the median time, that means the distribution is skewed to the left and would be appropriate for the distribution of histogram K. this is because when the distribution is skewed, the mean is pulled to the tail of the distribution and is either larger than or smaller than the median. In this case, for the mean to be smaller, the tail should be to the left.
The distribution of both histograms J and k combined will have a bimodal shape and the peaks would be around 0-8 and 40-48.
For a random sample of 36 staff members, the sampling distribution will follow a approximately normal distribution and have a mean of 25 minutes and a standard deviation of 18/ (square. root of 36) = 3 minutes. Because the sample size of 36 is larger than 30, it meets the CLT condition and can be assumed as approximately normal.
The test would be two sample t test for the difference of means1. Ho : MUs - MUstaff =01. Ha : MUs - MUstaff does not equal 0.1. MUs represents the true mean travel time for students and MUstaff represents the true mean travel time for staff members

Teacher Feedback

All of your answers are correct and give appropriate reasoning. A tip for part (a): when asked to make a choice in AP stats, you should explain not only why your choice is correct, but why the other option is incorrect. You could add “since histogram J is right skewed, it does not match the description” or something similar. All of your other answers are thorough, in context, and show strong understanding. Well done!

Student Response 3

If the mean is less than the median that means the data is skewed to the left, because the mean is affected by smaller values bringing it down. Histogram K appears skewed to the left, so it must represent the group of students who walk to school.
The distribution can be described as bimodal.
The shape of the histogram is approximately normal, because 36 is greater than 30, which satisfies the central limit theorem. The center is the mean, 25 minutes. The spread is the standard error, 18/sqrt36, or 3.
Two sample means t test1. H0: mu_students = mu_staff1. HA: mu_students is not equal to mu_staff1. where mu_students is the true mean travel time for all students at the school and mu_staff is the true mean travel time for all staff members at the school. Conditions:

The students and staff members were randomly selected.
Because the samples were randomly selected and the sample sizes are large (n_students = 50 and n_staff = 40), the sampling distribution of x̄_students - x̄_staff is approximately normal, so a two-sample t test for μ_students - μ_staff is appropriate.
There are at least 501 students and 401 staff members, so the 10% condition for independence is satisfied.

Teacher Feedback

Nice job! One small thing: for #2, you’re correct in calling the shape “bimodal,” but the rubric for a similar question only gave full credit for identifying where the clusters/peaks were centered. So you’d get partial credit. Full credit: “bimodal, with peaks at around 8 minutes and 40 minutes”

For part 4, your test name, hypotheses, and parameter definitions are correct. The conditions you included are statistically reasonable, but because the prompt only asked for the name of the test and the hypotheses, those conditions would not typically be required for credit on this part. On the AP exam, match your response to what the prompt specifically asks.

Quick FRQ Tips for This Kind of Question

For graph-description parts, don’t stop at one word like “skewed” or “bimodal” if you can be more specific.
For sampling distributions, include shape, center, and spread.
For inference, match your response to the task. If a prompt only asks you to name a procedure and state hypotheses, then give the correct test name, define the population parameters, and write H0 and Ha correctly.
If a prompt asks you to carry out inference, then include conditions, statistical work, and a conclusion in context. AP readers award credit for the components that the prompt specifically asks for.