On the left below, you'll find a copy of skeletal notes I used in MATH 110 for Fall 2015, and, on the right below, you'll find a copy of the skeletal notes I used in MATH 110 for Spring 2017. The lesson topic is Inference in Practice, which covers the details one needs to be aware of when practicing statistics in real life. This lesson is taught during the third unit of a 4 unit course. In the center, I describe what it is I am trying to achieve, and I discuss what I think did and did not work with these notes. The skeletal notes are also provided as pdfs (Fall 2015 and Spring 2017) so you may see what the students see.
MA 110 Fall 2015
Inference in Practice
Lesson Objectives:
To learn conditions necessary for inference
To understand the cautions of confidence intervals in practice
To understand the cautions of Tests of Significance in practice
To determine a sample size for desired margin of error in confidence intervals
To learn Type I and Type II errors
Statistical Inference methods:
These methods are also called
Recall,
We need the simple conditions to be true in order to trust these methods:
Note:
These procedures should not be used if
It's recommended to ask
1. ``Where did the data come from?"
e.g.
2. ``What is the shape of the population distribution?"
Ex 1) A professor is interested in how the 500 students in his class will rate today's lecture. He selects the first 20 students on his class list, reads the names at the beginning of the lecture, and asks them to go online to the course website after class and rate the lecture on a scale of 0 to 5. What is the reason why a confidence interval for the mean rating by all his students based on these data is of little use?
Cautions about Confidence Intervals:
Sampling distribution shows how a statistic varies in repeated random sampling.
This leads to a discussion of sample size:
Ex 2) Good weather, good tips? You read a newspaper article about the study that reports that with 95% confidence the mean % tip from restaurant patrons will be between 21.33 and 23.09 when the server writes the message on the bill stating the next day's weather will be good. Can you conclude that if you begin writing a message on patron' bills that the next day's weather will be good, approx 95% of the days you work your mean percentage tip will be between 21.33 and 23.09? Why or why not?
Cautions about Tests of Significance: 4 things --
How small a P is convincing?
Significance depends on H_a
Sample size influences statistical significance
Multiple analyses:
Ex 3) A researcher looking for evidence of extrasensory perception (ESP) tests 1000 subjects. Nine of these subjects do significantly better (P< 0.01) than random guessing.
(a) 9 seems like a lot, but you can't conclude these 9 have ESP. Why?
(b) What should researchers now do to test whether any of these 9 have ESP?
Power of Statistical Test:
For statistical tests, there are 4 scenarios:
H_0
H_0
H_0
H_0
Type I error:
Type II error:
While participating in the CoAT program, I created skeletal notes for the purpose of expanding the number of learning styles catered during my lectures.
The Lesson Objectives are meant to guide those students who like to know where we are headed in the lesson. (global)
I provide some notes already written for the students who like to listen to the instructor speak. (auditory)
In addition, I leave some notes blank for those who need notes to learn. (visual)
For this particular lesson, we have specific steps that students can use to guide them through real examples. (sequential)
The examples are for the students to work on their own and in their groups, respectively. (reflective and active)
I have questions and facts for students to explore and understand where do they fit in the conversation. (sensing)
This lesson connects to prior lessons on confidence intervals and tests of significance. (intuitive)
After the CoAT program, I believed I would make skeletal notes a requirement. However, based on comments by students, this did not seem a fair rule to enforce. Therefore, skeletal notes are still optional. Please see the CoAT analysis on skeletal notes for more detail. What worked and what did not work still hold true.
Changes in Notes from 2015 to 2017:
Clearly, the skeletal notes from Fall 2015 are much longer than Spring 2017. Each set of notes is meant to be covered in a class period of 55 minutes. The first set of skeletal notes would either take 2 class periods, which I do not like to do because of the disjoint effect, or I would need to speed through to cover all material. For the second set of skeletal notes, I decided I would cover less in more detail. The detail comes from class discussion and effective questioning. Students who use the skeletal notes are able to write down details from conversation.
Despite its shortened length, the notes from Spring 2017 contains more detail. For this particular topic, there is a lot to write. I believe students need to think, not just write. Therefore, I provide more notes so that students can think and listen.
A negative change may be that I have less exercises for students to work on in groups or alone. Notice, however, that in the first set of notes, the questions for the exercises are discussion based questions. These are the type of questions I will ask during the lecture in order to determine what students understand.
Effect on Students Learning:
This topic is significant, especially since students are required to do a semester project where they may be using inference procedures on real data they collect themselves. I highly recommend to students they need to read the chapter prior to class discussion.
Between Fall 2015 and Spring 2017, there seems to be a small increase in understanding based on this lesson alone. The project will further increase a student's understanding of inference in practice if they have a project that requires inference.
I am in favor of classroom activities that emphasize a particular lesson. In the future, I may require the reading and a response journal as a homework assignment. Then, on the next day, I will have an activity that supports what the student is supposed to have learned.
MA 110 Spring 2017
Inference in Practice
Lesson Objectives:
To modify conditions necessary for inference
To think about the cautions of confidence intervals
To determine a sample size for desired margin of error in confidence intervals
To think about Tests of Significance in practice
Recall the simple conditions are
SRS << population
variable of interest has N(mu, sigma)
sigma is known, but mu is not.
In practice, we may not have these conditions!!
Aside: we call these z-procedures because they use z-statistics and the standard Normal curve N(mu = 0, sigma = 1).
Where?
Inference is most reliable when data
If not true, conclusions can be challenged.
Common to apply inference to data not from random selection e.g.
Can we act as though
What?
Many basic s.i. methods require normality. BUT,
Often, we use life experience to assume
Or, we explore data.
Note: Ch 28 discuss methods that don't require normality. This is online. We may not have time to get to this :\
CONFIDENCE INTERVALS, mean +- z* (sigma)(sqrt{n})
We can have many different CI because we can have many different
The mean has sampling error. That is,
The CI does NOT take into account other variation. The margin of error only covers sampling error.
TESTS OF SIGNIFICANCE
Question: In your own words, state what the purpose of a test of significance is.
We ask:
How small is small enough?
How plausible is
Consequences
Different people will insist on
While it is common to
Significance depends on H_a: one-sided arguments are
5% significance level for n =
50 is different than 5% for n = 500
also depends on # of tests
When planning for a ToS, we need to take into consideration