DAREN STARNES: Hello, AP stat students.
And welcome to AP Daily practice sessions.
I'm Mr. Starnes coming to you from lovely Hilton
Head, South Carolina.
And this is session 8, where we're
focused on free response, the investigative task.
If you tuned in for session 7, that
was your first look at an investigative task question.
So this would be the second look, hopefully.
Reminder, if you haven't downloaded the questions
and tried them, please do that now.
You can click on the link at the beginning of the video.
You'll get a lot more out of this, if you've already
tried it and seen where the snags were
that you hit before you watch me do the solution.
So if you need to pause, now's the time.
All right, let's head on into the question.
This one says to compare success rates for treating allergies
at two clinics that specialize in treating allergy sufferers,
researchers selected random samples of patients
from the two clinics.
The following table summarizes the data.
So some keywords being underlined and doing
annotating that I like to do.
Strongly encourage you to do that on the exam
because our questions are so word intensive.
And you don't want to miss any of the important statistical
words or numbers in the problem.
Part A1, complete the following table
by recording the relative frequencies
of successful and unsuccessful treatments at each clinic.
So we're going to complete that table.
We're going to use relative frequencies.
And we're going to be sure we do the percent
or proportion of successful and unsuccessful treatments
at each clinic.
So let's start by focusing on clinic A. At clinic A,
successful and unsuccessful.
Well, we had 51 out of 139 unsuccessful.
And with my calculator, I got about 0.3669.
You could also give it as a percentage, 36.69%.
For the successful treatments, 88 out of 139 or about 0.633,
we don't worry about the exact number of decimal places.
So I'm using four here.
three would be fine.
Looking for clinic B?
What clinic B?
For the unsuccessful, we have 33 out of 68.
That's almost half, about 48.5%.
And for the successful, we've got 35 out of 68.
Just over half.
So about 0.5147.
So again, whether you do it in proportions
or whether you do it in percents, as long as
it's a relative frequency, not just the number
because that would be a frequency.
Part A2, based on the relative frequency
table in part A1, which clinic is more successful in treating
allergy sufferers?
Justify your answer.
Well, we can compare some proportions.
We just need to compare the right ones.
And we need to then justify based on our comparison.
So I'm going to look at successful treatment.
You could also look at unsuccessful treatment.
And think about that.
For the successful treatments, clinic A,
88 out of 139 or 0.63ish.
Clinic B, 35 out of 68 or 0.51ish.
So clinic A appears to be more successful in treating allergy
sufferers than clinic B. Why?
Because clinic A was successful for 63.3% of the allergy
sufferers it treated.
While clinic B was only successful
with 51.5% of the allergy sufferers it treated.
And I explicitly put which is greater than.
So it's an obvious comparison of which one is better
and which one's worse.
As I mentioned, you could have done unsuccessful treatment
and looked at the fact that clinic A had a smaller
proportion or percentage of unsuccessful treatments
than clinic B and done a similar solution that way.
So far, part A should not have surprised you too much.
These are the kinds of things you learn
to do earlier in the course.
You may not have done exactly this kind of question,
but there's nothing that's really a curveball yet.
Let's continue.
So we have the beginning part of the problem still with us.
Part B, based on the design of the study,
would a statistically significant result
allow the researchers to conclude
that receiving treatments at the clinic you selected
in part A2, that was clinic A by the way,
causes a higher percentage of successful treatments
than at the other clinic?
Explain your answer.
So remember, in A2, we said, with numerical evidence,
that clinic A appears to be more successful in treating allergy
sufferers.
The question is, if we get a statistically significant
difference between the clinics in terms
of their successful treatment rate,
can we conclude that the particular clinic selected
causes, I'm emphasizing that word when I say it,
a higher percentage of successful treatments?
Got your answer?
No.
We can't infer cause and effect from an observational study.
The patients here were not randomly assigned
with their allergies to go to one clinic or the other clinic.
If they had done a randomized experiment
and these were the data, then we could infer a cause and effect
conclusion, if the result was statistically significant.
But they didn't.
These are random samples of patient records.
The patients chose which clinic to go to.
So we can't establish cause and effect
from an observational study of this sort.
Also, there may be other variables besides just which
clinic they were treated at that affect the success
rates for treating allergies.
For example, maybe clinic A mostly treated mild allergies.
And so they did better treating them.
Whereas, clinic B maybe treated more severe allergy cases.
And that's what they're known for.
So there could be other variables
in the background that are helping
explain why clinic A would do better than clinic B.
But it's not because of a causal relationship between the clinic
they go to and the success or lack of success
of their treatment.
So B is about scope of inference based on the way
that the data were produced.
In this case, an observational study.
Part A was more about descriptive statistics
and data analysis.
So we've already gotten into two distinct content
areas in the course.
And we're going to just keep going.
So now, a physician who worked at both clinics
believed that it was important to separate
the patients in the study by the severity of the patient's
allergies, severe or mild.
The physician constructed the following interesting
looking mosaic plot.
The values in the mosaic plot represents the number
of patients who were either successfully treated
or unsuccessfully treated in each allergy severity
group within each clinic.
For example, the value, 78, represents the number
of patients treated in the mild group within clinic A.
Based on the mosaic plot, the physician
concluded the following.
For mild allergy sufferers, the physician
concluded that clinic B was more successful in treating
allergies.
And for severe allergy sufferers,
clinic B was also more successful
in treating allergies.
So they're looking at the relative proportions
or percentages in the mild bar.
You can see that clinic B does better.
And then they're looking at for severe allergies,
they're looking at the bars for severe.
And they're seeing that 10 out of 35 and 24 out of 56,
clinic B does better there too.
So where's the question?
Based on mosaic plot, we know what the physician concluded.
But for each clinic, we're asked, which allergy severity
is treated more successfully?
So we're just focused on individual clinics.
And we have to justify our answer.
So at clinic A, the mild allergies
are treated fairly successfully.
The success rate is 78 out of 104
compared to the severe allergies,
the success rate is quite a bit lower 10 out of 35.
And we can calculate those percentages.
75% for mild and about 29% for severe.
So clearly, mild is treated much more successfully.
In clinic B, well, over there, the mild allergies,
again, pretty successful.
11 out of 12 were successfully treated compared
to 24 out of 56 of the severe.
So clinic B was also more successful
at treating mild allergies.
And we can give the percentages or proportions there,
92% roughly success with the mild allergies.
Only about 43% with the severe.
OK, now let's pivot.
For C2, we're going to focus on which
allergy severity is more likely to be treated at each clinic?
Which allergy severity is more likely to be
treated at each clinic?
So at clinic A, if you look at the widths of these two bars
in the mosaic plot, the width is proportional to the number
of individuals in that group.
So in the mild group, there's 104 individuals.
And in the severe group, only 35.
So the mild bar is three times as wide, roughly.
So clinic A is more likely to treat people
with mild allergies is what that shows.
And we can document, in fact, it's about 75% roughly
and 25% for mild versus severe.
But you can actually see it from the width of the bars
and the mosaic plot.
So with that little tip, can you tell
for clinic B, which allergy severity is
more likely to be treated?
Well, I think I can.
How about severe?
Severe has got the wider bar.
There's 56 in that bar compared to only 12 over here.
So severe is the more common or likely to be treated at clinic
B. We can give some percentages.
Severe allergy sufferers are about 82ish% at clinic B.
And the mild allergy sufferers are only about 18% at clinic B.
So we can see which clinic treats more of one
type of allergy than the other.
We've also seen that both clinics are
more successful at treating mild allergy sufferers, which
makes sense.
Because mild allergies are probably easier
to treat wherever you go for your clinic.
And now, the finale of the investigative task
is to put all this together.
You might remember that the physician said some stuff.
And the physician said that clinic B
is more successful in treating both the severe
and the mild allergies.
So what we're going to do is use our answers from part C
to give a reason why the more successful clinic that we
identified in part A2, and that was clinic A,
is the same as or different from the physician's conclusion.
That clinic B is more successful in treating
both severe and mild allergies.
So just a reminder from the previous parts.
Here's what we said earlier about both clinics
treating more mild allergy sufferers than severe.
That was true at both clinics.
And then we subsequently said that clinic A
was more likely to treat the mild allergy sufferers.
Whereas, clinic B was more likely to treat severe.
So thinking about that, mild allergies
are easier to treat wherever you go.
But clinic A has a lot higher percentage
of mild allergy sufferers, which would make them look
better, more effective overall.
Clinic B has a lot more, percentagewise, severe allergy
sufferers.
And they're harder to treat.
So that's going to make clinic B look worse overall.
Which is why when we started, we just
looked at the overall two-way table.
We saw clinic A is being superior.
They have such a high success rate
with mild allergies at both clinics.
And clinic A had way higher percentage
of mild allergy sufferers.
So let's try to document that now.
So remember, what we said in A2, we said clinic A.
And that is different from the physician's conclusion
that it was clinic B that did better.
So we need to address that.
And then why does that happen?
At both clinics, the success rate
is higher for mild allergy sufferers.
75% versus 28.6% at clinic A. 92% versus about 43% at clinic
B for success in those different allergy suffering levels.
Now, the other twist is that clinic A
treats way more mild allergy sufferers
like 75% of its patients.
But clinic B treats a high percentage
of severe allergy sufferers, which
makes it look like their overall success isn't so good.
And it makes it look like clinic A has a really high success
rate because they are treating so many people
with mild allergies.
This interesting phenomenon, where
it looks one way to the physician,
but a different way to us because of the way
that the categories were combined in the two-way table
is known as Simpson's paradox just for your interest,
if you want to learn more about it.
So this is a case where an investigative task is taking
you beyond what you've learned in the course
to explore a phenomenon of statistical thinking
that you can figure out with the pieces
that they've scaffolded for you in the question.
Will every student figure it out?
Probably not.
But could you figure it out?
You could, if you've worked enough of these questions
to realize that you have everything you need from what
you've learned in the course and what's
provided in the question.
They really nicely scaffold it out for you.
And so if you get those first few parts under your belt,
gain some confidence.
Earn some points.
Then as you head into the back half of the question,
you're not as thrown off by the twists
that you know are coming.
That's how investigative tasks are designed on our exam.
And the students who do really well on them
show good statistical reasoning.
And they're rewarded with good scores on the free response
section and often on the exam, overall.
So remember, you can do these investigative tasks.
What's up next?
Well, we hope that you'll continue to prepare
for the AP statistics exam.
One way you can do that is to go onto AP Central-- sorry, AP
Classroom and look in the review tab for some of on demand
review videos that have completely different questions
from the ones you've seen in these practice sessions
to give you additional practice.
Practice is what's going to help you along
with the feedback we can give in these videos for you.
And in addition to that, you can work with your fellow students
in class and take the advice of your teacher
to make sure that you do as well as possible on the AP exam,
which we really hope for you.
So thanks so much for tuning in.
And sincere best wishes on the AP exam.