Research Lesson Report
Lesson Study Team Members: Jeff Baggett, Brooke Fridley, David Reineke
Department and Institution: Department of Mathematics at University of Wisconsin –
La Crosse
Lesson Topic: Interpretation of Confidence Intervals
Course: Elementary Statistics (MTH 145)
Course Level: Introductory
Class size: 40 students
Length of Lesson: 1 class period for original lesson study and 2 class periods for revised
lesson study.
Classroom setting: classroom
BACKGROUND:
Our lesson study will focus on the Elementary Statistics (MTH 145) course. The course
introduces students to the basic ideas of probability and statistics. Approximately 1200
students take the course each year, with students ranging in majors and years on campus.
The course meets a general education requirement, with some students needing MTH 145
for their major. All three of the lesson study participants teach the course on a regular
basis.
Our group decided to address the topic of confidence intervals and the interpretation and
use of confidence intervals, in particular focusing on the interpretation of confidence
intervals (i.e. what intervals say and what they do not say). We chose this topic after
having noticed students having difficulty with the interpretation and use of the confidence
interval and not so much the computation of the confidence interval.
LESSON DESIGN:
The learning goals are that by the end of the lesson, students will understand the purpose
of a confidence interval, how to calculate a confidence interval and how to interpret a
confidence interval correctly in the context of the problem. With the completion of the
lesson, we hope to address and prevent common misconceptions and misinterpretations
of confidence intervals.
The overarching purpose of the lesson is building on the basic idea of statistical
inference. By the end of the entire course, the hope is that students have acquired an
understanding of statistical inference (i.e. making a claim about the population from
information collected via a random sample or randomized experiment) and to gain insight
into the idea of sampling variability.
The plan for the study was to design a lesson study that would introduce students to the
idea of a confidence interval and how the computed confidence interval can be used to
give information about the population parameter (here the population mean). In designing
the lesson study, we wanted to emphasis how confidence intervals for the population
mean are interpreted and how they should NOT be use. For example, it is common for
students to believe there is 95% chance that the population mean falls in the interval or
that 95% of all elements in the population fall in the confidence interval.
General overview of lesson:
1. Pre-test on the meaning/interpretation of a confidence interval for the mean. (2-3
minutes)
2. Discussion on the derivation of a confidence interval for the population mean. (15
- 20 minutes)
3. Group worksheet investigating the computation of a confidence interval and the
subsequent interpretation of the confidence interval. Below is a copy of the
worksheet. Note: Every student has a different sample of size 30 pennies. (15
minutes)
MTH 145 – Elementary Statistics
Confidence Interval Worksheet
We’re going to use confidence intervals to estimate the true average age of a penny in the jar.
Our population of pennies has σ = 12.62 years.
A simple random sample of 30 pennies from the jar gives the following ages in years:
30,40,22,25,21,30,14,14,40,5,5,29,26,34,2,10,13,9,3,3,53,36,20,5,21,3,3,8,11,30
For this sample of pennies, the average age is
x = 18.83.
The formula for the confidence interval for estimating µ is
x + z* (σ /
n ),
where z* is the critical value for a level C confidence interval as explained in class.
1. Find a 68% confidence interval for the true average age of a penny (use z* = 1).
2. Find a 95% confidence interval for the true average age of a penny (use z* = 2).
3. How confident are you that the 95% confidence interval contains the true average age of
a penny?
4. What is the probability that your 95% confidence interval contains the true average age of
a penny?
5.
If the true average age is known to be µ = 19.92 years. Does your 68% CI contain µ?
Does your 95% CI contain µ?
4. Discussion of the activity and the answers students gave to the last three questions
on the worksheet. In doing so, we discussed that you are not able to apply
probability statements to the confidence interval and that the confidence interval
for µ tells us nothing about individuals in the population, only about the
population mean. (10 minutes)
5. Made the following table on the chalkboard and asked students to raise their hand
if their computed intervals contained µ. Roughly 68% of the classes 68% CI
contained µ while roughly 95% of the classes 95% CI contained µ. We then
discussed the impact of the confidence level on the confidence interval. (5
minutes)
Interval contains µ Interval does not contain µ
68%
95%
6. Post - test on the meaning/interpretation of a confidence interval for the mean. (3
minutes)
The lesson study was developed to have students think about what a confidence interval
for the population mean tells in the context of a problem (such as estimating the average
age of pennies in the jar). In having the student answer various questions on their own
during the activity, we were able as a class to discuss the various answers (correct and
incorrect) to the questions on the worksheet. In the lesson study, students also
investigated the meaning of a 68% CI and a 95% CI and the impact of the confidence
level on the performance of a CI.
We predict that the activity will not only help students with the computation of a CI for
various confidence levels, but also with the interpretation of the constructed confidence
interval. With the interpretation of the CI being difficult (in comparison to the
computation of the CI), we do not think this activity alone will dispel all
misinterpretation of CI’s, but will help students to be careful about how they interpret
CI’s. We believe the activity will be very successful in the objective that students will be
able to compute the specified CI.
As for student interest in the lesson plan, we feel that the use of an activity will keep
students engaged in the topic of confidence intervals more than a traditional lesson which
consist of an entire hour of lecture.
FINDINGS:
A pre-test and post-test on interpreting confidence intervals will be given and collected
from each student. The pre and post quiz is presented below.
MTH 145: Introductory Statistics
Pre-Quiz: Confidence Interval Interpretation
Name ________________
Put a check in the box preceding the correct interpretation of the confidence interval for the mean.
Confidence Interval: ($1,835, $2,691)



With 95% confidence, the average income tax refund for Wisconsin households is
between $1,835 and $2,691.
The probability that the average income tax refund for Wisconsin households falls
between $1,835 and $2,691 is 0.95.
95% of all households in Wisconsin will have a average income tax refund between
$1,835 and $2,691.
MTH 145: Introductory Statistics
Post-Quiz: Confidence Interval Interpretation
Name________________
Put a check in the box preceding the correct interpretation of the confidence interval for the mean.
Confidence Interval: (8.3 minutes, 16.7 minutes)



90% of all airplanes at Chicago’s O’Hare International Airport take between 8.3 and 16.7
minutes to taxi from the gate to the runway.
We can be 90% sure that airplanes at Chicago’s O’Hare International Airport take
between 8.3 and 16.7 minutes to taxi from the gate to the runway, on average.
There is a 90% chance that the mean taxi time for airplanes at Chicago’s O’Hare
International Airport falls between 8.3 and 16.7 minutes.
While in the pre-quiz, the number of students answering correctly was higher for the
correct answer than for the two decoys, there was a high number of students choose the
decoys.
Correct: 28
Decoy 1 (interpreting for individuals): 17
Decoy 2 (interpreting with a probability statement): 18
At the end of the lesson, a post-quiz was given in which about the same number of
students gave the correct answer while the remaining students (except for 1 student)
chose Decoy 2 (interpreting the interval with a probability statement). We clear up one
misconception of the interpretation of confidence intervals in that the interval for the
population mean (parameter) tells us nothing about individual responses. That is, a 95%
confidence interval for the mean does not tell us that 95% of observations for individuals
in the population fall in the confidence interval.
Correct: 31
Decoy 1: 1
Decoy 2: 32
Note that one student did not turn in their pre-quiz.
Our initial lesson study plan did not have enough discussion time to allow for enough
observable student activity. That is, there was not enough student discussion time for us
to observe student comments/discussion of the interpretation of confidence interval. We
were not able to observe students’ ability to carry out the computation in the construction
of a confidence interval (which is generally not the difficult portion of understanding
confidence intervals). A pattern noted in student performance is that students tend to
latch on to key words rather than key concepts. As a result, some students fail to
recognize a concept when it is expressed using slightly different (but equivalent)
terminology.
CONCLUSIONS:
Based on the results of the pre and post quiz, the misconception regarding the
interpretation of a confidence interval by applying it to individuals rather than means
seems to have been adequately addressed by the lesson. However, students still do not
have a clear understanding of the interpretation of a confidence interval for the mean as it
relates to the subtle difference between probability and confidence.
During the lesson, students certainly learned the mechanics of carrying out the
computations necessary to construct a confidence interval, but this was not our focus and
is rarely a problem. By providing students with a scenario in which random samples had
been drawn from a population, we hoped to make use of the foundational concept of the
sampling distribution of the mean, which is covered just prior to confidence intervals.
This is a point at which many students get confused or just completely miss the
association with the sampling distribution of the mean.
Some students picked up on the correct interpretation of confidence intervals during the
series of questions that takes place after the students have constructed confidence
intervals for different samples taken from the same population. As students raised their
hands to indicate whether or not their particular interval contained the population mean,
they could look around the room to see that sometimes intervals capture the population
mean and sometimes they do not, and that that process is governed by the level of
confidence. Furthermore, when students are asked about their specific interval, the issues
of probability and confidence can somewhat be distinguished.
RECOMMENDATIONS:
We did not have enough discussion of confidence interval interpretation in different
settings, using different terminology, or demonstrations that illustrate the difference
between probability and confidence. Based on our analysis, we would change the lesson
study to a two day lesson rather than a one day lesson, in which the mechanics of
constructing a confidence interval is covered on the first day and the interpretation of the
confidence interval on the second day.
We are also going to include a time for students to discuss (and teachers to observe) the
answers to the pre-quiz given the day before and why one answer was correct and the
other two answers incorrect. In addition to these changes, we are going to include a short
demonstration drawing out the difference between confidence and probability before and
after a coin toss and/or other events. Lastly, we will add more questions that will lead to
class discussion to the worksheet.