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


  • 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.

    NCTM 2016 San Francisco


  • 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


  • So Where Do We Start??

     In this session, we will consider class activities/demonstrations (depending on your access to technology) that address

     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


    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


    Observational Studies and Sample Surveys

    Question of interest: How far off might my estimate


     Experiments

    Question of interest: Could this have happened by

    chance when there is no difference in the response to

    the different experimental conditions?

    NCTM 2016 San Francisco


    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


     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.

    NCTM 2016 San Francisco


  • 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


     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


     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


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  • 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

    about 0.07 of the actual proportion of adults who would

    choose the man on the left as Tim.

    NCTM 2016 San Francisco


  • Motivating Margin of Error

     Extension—If you have access to technology, have students each

    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.

    NCTM 2016 San Francisco