Name: _________________
AP Statistics Handout: Lesson 8.3
Topics: two-sample t-interval for a difference of means
Lesson 8.3 Guided Notes
Two Facts about College:
1. On average, college graduates make ___________________________ than high school
graduates. ($30,000 more per year, in 2018)
2. ________________________________ of college, the average economic return to a college
degree is still _____________________________________________.
Are college and income causally related?
Causation
Correlation
For “margin” students (those on the
fence of attending college based on
motivation and qualifications)…
• If college effect is causal: Educators
should encourage margin students
to attend college
• If college effect is correlation:
Educators shouldn’t always
encourage margin students to
attend college
Florida College Study*
A Yale economist used a regression discontinuity (a “quasi-experimental” method) to explore whether
college has a causal impact on wages. AP Stats doesn’t cover quasi-experimental methods, but they’re
really cool. We’ll analyze a modified version of this study using a concept we do learn in AP Stats: the
two-sample t-interval. In the space below, describe how the Florida College Study obtained its two
comparison samples. Then, describe why their methods allow for a causal comparison:
*Study: Zimmerman, Seth. “The Returns to College Admission for
Academically Marginal Students.” Journal of Labor Economics, 2014,
vol. 32 (4). https://www.jstor.org/stable/10.1086/676661?seq=1
Material adapted from the Skew The Script curriculum (skewthescript.org)
Lessons made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
2
Technical notes for technical folks:
• Regression Discontinuity uses multiple regression, which isn’t covered in AP Stats. We’re doing a
modified version with a two-sample t-interval (which is covered in AP Stats)
• Technically, we’re not really doing regression discontinuity. But what we’re doing uses similar
logic and reflects the paper’s main outcomes.
• We modified the data to make sure our results (including standard errors) would be similar to
Zimmerman’s (specifically, his IV effect estimates for students with GPA’s within 0.15 points of
the cutoff - see Table 5 in his paper – using annual rather than quarterly earnings as the
outcome).
Study results, for students near the GPA cutoff:
Mean Income
Stdev. Income
n
College
$35,764
$27,147
202
High School
$28,964
$21,899
190
“Income” is students’ annual income 8-14
years after high school (in 2005 dollars)
If college made a difference for margin
students, just-above cutoff students should
have a ________________________________
than just-below cutoff students.
Show calculations for 𝑥̅𝐶 − 𝑥̅𝐻𝑆 :
Could the difference in outcome have happened by chance alone? Or did college admission actually
raise wages? Let’s make a ____________________________ to see if difference between these means is
significantly large.
Two-sample t-interval for a difference of means
Hypotheses: We don’t need to set up hypotheses to construct a confidence interval, but doing so is going
to help us conceptualize why our interval is useful.
𝐻0 : 𝜇𝐶 = 𝜇𝐻𝑆
𝐻𝐴 : 𝜇𝐶 > 𝜇𝐻𝑆
The null (______________) hypothesis: there is ___________________
in average wage between the college and high school groups. You’re
seeing if there’s evidence to reject this claim.
The alternative (________________) hypothesis: the college group
_______________________ (on average) than the high school group
Rewrite these hypotheses in a more mathematically convenient way:
Where:
μC is the mean earnings among _____ college students (just above GPA cutoff)
μHS is the mean earnings among _____ high school students (just below GPA cutoff)
Material adapted from the Skew The Script curriculum (skewthescript.org)
Lessons made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
3
Making the interval
Mean Income
Stdev. Income
n
College
$35,764
$27,147
202
High School
$28,964
$21,899
190
Under certain conditions:
𝜎 2
~ Norm(𝜇 = 𝜇𝐶 − 𝜇𝐻𝑆 , 𝜎 = √ 𝑛𝑐 +
𝑐
𝜎𝐻𝑆 2
𝑛𝐻𝑆
1. Show the steps we take to get our t-distribution (and its final parameters):
2. Show the steps for calculating the final confidence interval (and write down the interval):
3. Interpret your final interval and comment on whether 0 is inside or outside your interval (and why
that’s important):
Material adapted from the Skew The Script curriculum (skewthescript.org)
Lessons made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
)
4
The Four Step Process for Inference
In the Florida College Study, two “as good as randomly” assigned groups of students (with similar GPA’s)
were given admission to college or were denied. Summary statistics are provided.
a) Construct and interpret a 95% confidence interval for the difference in mean incomes between
marginal students who attend and do not attend college.
b) Use your interval to draw a conclusion about whether college creates an “income boost” for marginal
students.
For “DO” phase: Calculator steps for two-sample t-interval for means:
STAT → TESTS →
0: 2-SampTInt…
For groups 1&2:
𝑥̅ : sample mean
𝑠𝑥 : sample stdev.
n: sample size
C-lvl: Confidence Lvl
Pooled → NO
Material adapted from the Skew The Script curriculum (skewthescript.org)
Lessons made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)
Output
5
Lesson 8.3 Discussion
Confidence interval: $1,906 to $11,693
Interval shows us that the “college boost” is statistically significant, but is it practically important?
• Study author (Zimmerman) shows, even with rising college costs, this “college boost” represents
a substantial net gain over time.
• So, yes, it’s practically important! College “pays off” for students at the margin of admission.
Policy Implications
Many colleges have tried to recruit more students from underrepresented backgrounds (e.g. students of
color and low-income students).
• One problem: Because students from these backgrounds disproportionately attend lowerperforming schools, many have lower GPA’s and academic qualifications.
• Colleges worry underqualified students will have bad outcomes (not graduate, student debt,
etc.)
Discussion: To help recruit students from a wider array of backgrounds, Universities in Florida are
considering lowering their admissions standard for high school GPA’s. Would this be a wise move?
Explain your answer using the Florida College Study results.
Lesson 8.3 Practice
Teachers: provide exercises from your AP Stats textbook (or other resources) about the content covered
in this lesson. This lesson matches with the following sections of the most widely-used AP Stats
textbooks:
•
•
•
•
The Practice of Statistics, 4th-6th editions (+CED-aligned 6th edition update): section 10.2
Stats: Modeling the World, 4th/5th editions: ch 23, 3rd edition: ch 24
Statistics: Learning from Data, 2nd edition: sections 13.3-13.4
Advanced High School Statistics, section 7.3
Material adapted from the Skew The Script curriculum (skewthescript.org)
Lessons made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License
(https://creativecommons.org/licenses/by-nc-sa/4.0)