# Making Inferences and Justifying Conclusions Making Inferences and Justifying Conclusions Roxy Peck

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• Making Inferences and

Justifying Conclusions

Roxy Peck

Cal Poly, San Luis Obispo

NCTM 2016 San Francisco

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• Common Core State Standard in Mathematics

 S-IC 3 Recognize the purposes of and difference among sample surveys, experiments, and observational studies; explain how randomization relates to each.

 S-IC 4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

 S-IC 5 Use data from a randomized experiment to compare two treatments; use simulation to decide if difference between parameters are significant.

 S-IC 6 Evaluate reports based on data.

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• Common Core State Standard in Mathematics

 These standards include difficult (but important)

statistical concepts.

Concepts of random selection, random assignment,

study design, sampling variability, margin of error,

statistical significance are not just for AP Statistics

anymore! They are now part of the “for all” part of the

high school curriculum.

 In most Common Core schools (and “Common Core like

schools”), every high school teacher of mathematics is

now being asked to develop students’ statistical thinking

as well as their mathematical thinking.

 This is a big challenge! NCTM 2016 San Francisco

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• So Where Do We Start??

 The difference between observational studies and experiments.

 The difference between random selection and random assignment.

 How study design relates to the types of conclusions that can be drawn.

 Using simulation to develop the concept of margin of error.

 Using simulation to develop the concept of statistical significance.

 But we won’t have time, so will go very quickly through the first three and then focus on the last two.

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• Observational Studies versus Experiments

 Observational study

 Observe characteristics of a sample selected from one or more populations

 Goal is to use sample data to learn about the corresponding population

 Important that the sample be representative of the population

 Experiment

 Study how a response variable behaves under different experimental conditions

 Person conducting the experiment decides what the experimental conditions will

be and who will be in each experimental group

 Important to have comparable experimental groups

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• Observational Studies versus Experiments

 Observational studies (includes sample surveys)

 Want random selection from population of interest since it is important to have a

sample that is representative of the population.

 Random selection enables generalizing from sample to the population.

 Experiments

 Want random assignment of “subjects” to experimental conditions to create

comparable experimental groups.

 Random assignment enables drawing a cause and effect conclusion (changes

in the experimental conditions cause change in response).

 Experiments may or may not include random selection of subjects.

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• So What is Randomization??

Random selection

Random assignment

Randomization

Let’s keep it simple and not confuse students!

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• Give students lots of practice doing

things like this…

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• Margin of Error and Statistical

Significance

Observational Studies and Sample Surveys

Question of interest: How far off might my estimate

be?

 Experiments

Question of interest: Could this have happened by

chance when there is no difference in the response to

the different experimental conditions?

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Margin of Error

Statistical Significance

CCSS limits these concepts Margin of errorobservational studies

Statistical significanceexperiments

• Using Simulation to Develop Concept

of Margin of Error

How far off might my estimate be?

Study on facial stereotyping (thanks to Allan

Rossman and Beth Chance for this example).

Reference: Psychonomic Bulletin & Review, 2007

14(5), 901-907.

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• Bob or Tim?

One of these men is named Bob and one is named Tim.

They were asked “Which man is named Tim and which is

named Bob?”

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• Bob or Tim?

 Want to use data to estimate the proportion of U.S. adults that

would choose the man on the left as Tim.

 We will assume that it is reasonable to assume that this group is

representative of the population of adults in the U.S.

 For this group, the proportion who chose the man of the left as Tim

is:

 But I don’t have an internet connection so I am going to pretend

that we are a group of 100 people and that 78 picked the man on

the left as Tim. With a class where I would have internet access, I

would use the real class data. The proportion who choose the man

on the left as Tim is pretty consistently around 0.80.

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• Motivating Margin of Error

 Based on my sample of 100 people, my estimate of the proportion

of U.S. adults who would choose the man on the left as Tim is 0.78.

 But I don’t expect this to be exactly equal to the actual population

proportion. How close can I expect my estimate to be to the actual

value?

 Margin of error is the maximum likely error. It would not be likely that

my estimate would be off by more than this amount. “Likely” is

defined in terms of 95%--If I were to takes samples from the

population and use each sample to estimate the population value, 95% of these estimates would differ from the actual value by less

than this amount.

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• Motivating Margin of Error

 How do we get a sense of how far off my estimate is likely to be?

 Create a BIG hypothetical population with a proportion of “successes”

that is equal to my sample proportion.

 Take a random sample of the same size as my original sample from the

big hypothetical population and calculate the proportion for this

simulated sample.

 Repeat many times to get a collection of simulated sample proportions.

 Look at the simulated sample proportions to see how far off they

tended to be from the known proportion for my BIG hypothetical

population.

 The margin of error based on the simulated sample proportions is a

reasonable estimate of the margin of error I should associate with my

original estimate.

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• Motivating Margin of Error

 http://www.rossmanchance.com/ISIapplets.html

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http://www.rossmanchance.com/ISIapplets.html

• Motivating Margin of Error

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• Motivating Margin of Error

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• Motivating Margin of Error

 95% of simulated sample proportions were between 0.71

and 0.84. Since the actual proportion in the BIG

hypothetical population was 0.78, we could say that

about 95% of the simulated sample proportions were

within about 0.07 of the actual population value.

Margin of error is 0.07.

 So we think that our estimate of 0.78 is probably within

choose the man on the left as Tim.

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• Motivating Margin of Error

carry out a simulation to get their own margin of error estimates. Compare with other students so that they see that the simulation

method tends to produce consistent results.

 Also works for simulating margin of error for estimating a population

mean. But to create the BIG hypothetical population to sample

from, we create a population that consists of a large number of

copies of our sample (which we think is representative of the

population). Sampling from this BIG hypothetical population is equivalent to sampling with replacement from the original sample.

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