Requisite Knowledge for Teachers, Assessment Questions, Technology Allan Rossman, Soma Roy, Beth...

Requisite Knowledge for Teachers, Assessment Questions, Technology

Allan Rossman, Soma Roy, Beth ChanceDept of Statistics, Cal Poly – San Luis [email protected]@[email protected]

Cal Poly

STAT 217, 221 Algebra-based, wide variety of majors Several instructors (Chance, McGaughey, Rossman,

Roy) use simulation/randomization-based curriculum Curricular materials (ISI) developed with Tintle, Cobb,

Swanson, VanderStoep (Wiley, to appear) STAT 301

Calculus-based, mostly Math and Stat majors Use simulation/randomization-based ISCAM

materials (Chance and Rossman, 2nd ed.)

JMM 2013 2

Question on faculty interviews Most challenging topic to teach in Stat 101?

“Right” answer: Sampling distributions Central to understanding statistical inference:

What would happen with repeated random sampling? Very difficult cognitive step for students:

Seeing proportion or average not as a number but as a (random) variable with a distribution

Equally “right” answer: Randomization distributions Analogous to sampling distributions:

What would happen with repeated random assignment? Many teachers have not studied

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1. What is a randomization distribution? Sleep deprivation study

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ITS/CSS

need to show one of the mixed up distributions

Challenges in understanding this graph

What’s the variable? Difference in group means

What are the observational units? Random assignments

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What to look for in this graph? Shape

We really don’t care Only relevant to justify using a theoretical

approximation (t) Center

We really don’t care Only relevant to confirm that we correctly

simulated under the null (0)

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What to look for? (cont)

Variability This we care about This tells us what values of the statistic are

typical, what values are unusual, what values are rare when the null model is true

Seeing where observed value of statistic falls in the randomization (null) distribution determines p-value, strength of evidence

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Exact randomization distribution Distribution of differences in group means for

all 352,716 possible random assignments

Appropriate for calculus-based class

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2. Why simulate?

Study: 14 of 16 infants chose “nice” over “mean” toy

Two possible explanations Infants have genuine preference for nice toy Infants choose randomly

Why simulate? To investigate what could have happened by

chance alone (random choices), and so … To assess plausibility of “randomly choose”

hypothesis

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Why simulate? (cont)

Non-trivial for students to understand, articulate motivation for simulation Especially understanding that simulation is conducted

assuming null to be true, in order to assess plausibility Some students think that simulation means

replicating the study, generating a larger sample Some students think any use of software (e.g.,

to calculate a t-statistic) constitutes a simulation

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3. Four more examples

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A:

B:

C:

D:

Praised for intelligence Praised for effortLied 11 4

Did not lie 18 26Total 29 30

Gilbert on shift Gilbert not on shiftPatient death 40 34

No death 217 1350Total 257 1384

Year 1950 Year 2000Native born 219 258

Born elsewhere 281 242Total 500 500

Male FemaleDonated blood 126 104Did not donate 497 609

Total 623 713

Four more examples (cont)

Identical structure Two binary variables, 2×2 table of counts Could apply two-proportion z-test, chi-square test

But there’s a very important difference …

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Very different uses of randomness A: Random assignment B: Independent random samples C: One random sample D: No randomness

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Total 623 713


No death 217 1350Total 257 1384


Very different scope of conclusions to be drawn A: Cause/effect B: Generalize to two popns C: Generalize to one popn D: Rule out “random chance”

explanation

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Total 623 713


No death 217 1350Total 257 1384

Four more examples: Key question Should inference method mimic the

randomness in data collection? A: Randomization test B: Random sample/bootstrap with fixed margin C: Random sample/bootstrap with only total fixed D: ???

Permutation test

I’ve answered yes for calc-based course, no for algebra-based course But I think instructors should be aware of issue

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4. New example, three approaches Halloween treat study:

148 chose candy, 135 chose toy Two-sided test of equal likeliness Three approaches

Simulation-based approximation Normal-based approximation Exact binomial calculation

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Simulation-based approximation

Produces various approximate p-values

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Normal-based z-test

Produces one approximate p-value: .4396

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Exact binomial p-value

Produces one exact answer: .4758

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Simulation vs. theory vs. exact Expect algebra-based students to understand

simulation and theory Maybe it’s enough to expect them to select, apply

any relevant test procedure for one-proportion setting But expect calculus-based students to

understand relationships among 3 approaches Notice that simulation “beats” theory here

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5. How to estimate parameter? Do not reject the value .5 for parameter in

Halloween treat example What other potential values of parameter

would not be rejected? Perform more tests, using either simulation or

normal approximation or exact binomial Using some pre-specified significance level Define as plausible the values not rejected

Confidence interval for parameter: values not rejected by test

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Summary: What teachers need to know Randomization distribution Motivation for simulation

Common student misconceptions Connections among data collection,

inference procedure, scope of conclusions Subtle distinctions among simulation methods

Simulation vs. theory vs. exact Confidence interval as inversion of test;

interval of plausible values

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Assessment:Our Favorite Questions – on homework, quizzes, project reports, exams Describe how to use simulation/randomization to calculate

a p-value

Interpret a p-value in the context of the study

State an appropriate conclusion in the context of the study

Note: You are welcome to use any or all of the example questions. If you do, we would be thrilled if you could share your results with us. Thanks!

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Describe how to use simulation to calculate a p-value Example 1: “A research question of interest is whether

financial incentives can improve performance. Alicia designed a study to test whether video game players are more likely to win on a certain video game when offered a $5 incentive compared to when simply told to “do your best.” Forty subjects were randomly assigned to one of two groups, with one group being offered $5 for a win and the other group simply being told to “do your best.” She collected the following data from her study:

Explain in detail how you would conduct a simulation (say with pennies or dice or index cards) to obtain a p-value for this research question.”

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$5 incentive “Do your best” Total

Win 16 8 24

Lose 4 12 16

Total 20 20 40

24

Describe how to use simulation to calculate a p-value(contd.) What we look for in the answer:

Simulation assumes null is true Simulation mimics the randomness of the study design

(depending on course) Records the value of an appropriate statistic for many

repetitions Compares value of observed statistic to simulated

randomization distribution, to calculate the proportion of simulated values of statistic at least as extreme than the observed statistic


Interpret a p-value Example 2: In an article published in Psychology of Music

(2010), researchers reported the results of a study conducted to investigate the effects of “romantic lyrics on compliance with a courtship request.” … When a participant came in for the study, she was randomly assigned to listen to either a romantic song or a neutral song. After three minutes, she was greeted by a male “confederate” … who … asked for her phone number so that he could call her up to ask her out.

Of the 44 women who listened to the romantic song, 23 gave their phone numbers, whereas of the 43 who listened to the neutral song, only 12 did.

The p-value is computed to be 0.0103. Interpret this p-value in the context of the study.


Interpret a p-value (contd.) What we look for in the answer:

Similar to what look for when describing simulation What the p-value is a probability of - four components

Assuming the null hypothesis is true Randomness due to study design Compares observed statistic to simulated randomization

distribution Direction (as or more extreme)

All in context


State an appropriate conclusion

Example 3: To investigate whether there is an association between happiness and income level, we will use data from the 2002 General Social Survey (GSS), cross-classifying a person’s perceived happiness with their family income level. The GSS is a survey of randomly selected U.S. adults who are not institutionalized.

The p-value is found to be < 0.001.

State your conclusion in the context of the study.


State an appropriate conclusion (contd.) What we check that the answer addresses, (where applicable) and includes appropriate justification for (S) Statistical significance (E) Estimation, i.e, statistical confidence (C) Causation (was random assignment used?) (G) Generalizability (whom does the sample represent?)

All in context Student response:


More Examples


Example 4 (from AP Statistics): New statistic = mean / median. Give simulation results for values of mean / median, based

on a normal population

Give the observed value of mean / median for given sample data.

Do the simulation results suggest that the underlying population is skewed to the right? Explain.


Example 5: Present a graph of the null distribution and ask At what number should the graph center? Why? Shade the region of the graph that represents the p-

value. Calculate the approximate p-value. Which aspect of the graph tells you whether the study

results are statistically significant: (a) Shape, (b) Center, or (c) Variability?


For example,

Do incentives work? …A national sample of 735 households was randomly selected, … 368 households were randomly assigned to receive a monetary incentive along with the advance letter, and the other 367 households were assigned to receive only the advance letter; 286 in the incentive and 245 in the no-incentive group responded to the telephone survey.

Context: Difference in proportions


• Center makes sense? Why?• Shade region denoting p-value.• Are the results from the study

statistically significant? How are you deciding? (a) Shape, (b) Center, or (c) Variability

Example 6: Present 2+ graphs and ask which represents the null distribution, where

One graph centers at the hypothesized value One graph centers at the observed statistic Perhaps include a third graph that centers at some

commonly used null value (for e.g. 0.5 for proportions, and 0 for difference in means) which is not the correct null value for current study.


For example, In playing Rock-Paper-Scissors against the instructor, 8 out of 42 students picked scissors. To test whether these data provide evidence that such players tend to pick scissors less often than would be expected by random chance, which graph (A, B, or C) is the appropriate null distribution?


Center @ 14 (1/3 of 42)

Center @ 21 (1/2 of 42)

Center @ 8(observed count)

Example 7: Consider the following output.

Make up a research question for which this is plausible output. Clearly specify what the observational units and variable(s) in the study are. Do not use any of the contexts discussed in your lecture notes, labs, assignments, or any practice material.

Goal of question: Can students correctly identify the setting? How many variables and what type of variables? Direction of alternative (one-sided vs. two-sided)?


Multiple Choice Example 8:

Randomized experiment Could the difference in sample means (18) have occurred

by chance alone? Describes a tactile (card shuffling) simulation process and

gives the following graph based on 1000 repetitions

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Sample size

Mean score

$5 incentive 20 98“do your best” 20 80

37

Example 8 (contd.)

What does the histogram tell you about whether $5 incentives are effective in improving performance on the video game?

a) The incentive is not effective because the distribution of differences generated is centered at 0.

b) The incentive is effective because distribution of differences generated is centered at 0.

c) The incentive is not effective because the p-value is greater than .05.

d) The incentive is effective because the p-value is less than .05.


Example 8 contd.

1. Explanation for the simulation process?

a) Allows to determine whether the normal distribution fits the data.

b) Allows to compare actual result to what could have happened by chance if gamers’ performances were not affected by treatment

c) Allows to determine the % of time the $5 incentive strategy would outperform the “do your best” strategy for all possible scenarios.

d) Allows to determine how many times she needs to replicate the experiment for valid results.


Example 8 (contd.)

2. Which of the following was used as a basis for simulating the data 1000 times?

a) The $5 incentive is more effective than verbal encouragement for improving performance.

b) The $5 incentive and verbal encouragement are equally effective at improving performance.

c) Verbal encouragement is more effective than a $5 incentive for improving performance.

d) Both (a) and (b) but not (c).


Example 8 contd.

3. Approximate p-value in this situation? Recall that the research question believes that the incentive improves performance.

a) 0.220 divided by 100 instead of 1000

b) 0.047 2-sided p-value instead of 1-sided

c) 0.022 correct answer

d) 0.001 plain wrong


Example 8 (contd.)

5. Which of the following is the appropriate interpretation of the p-value?

a) The p-value is the probability that the $5 incentive is not really helpful.

b) The p-value is the probability that the $5 incentive is really helpful.

c) The p-value is the probability that she would get a result as extreme as she actually found, if the $5 incentive is really not helpful.

d) The p-value is the probability that a student wins on the video game.


Example 9: You want to investigate a claim that women are more likely than men to dream in color. You take a random sample of men and a random sample of women (in your community) and ask whether they dream in color, and compare the proportions of each gender that dream in color.

1) If the difference in the proportions (who dream in color) between the two samples turns out not to be statistically significant, which of the following is the best conclusion to draw? (Circle one.)

a) You have found strong evidence that there is no difference between the proportions of men and women in your community that dream in color.

b) You have not found enough evidence to conclude that there is a difference between the proportions of men and women in your community that dream in color

c) Because the result is not significant, the study does not support any conclusion


Example 9 (contd.):

2) If the difference in the proportions (who dream in color) between the two samples does turn out to have a small p-value, which one of the following is the best interpretation? (Circle one.)

a) It would not be very surprising to obtain the observed sample results if there is really no difference between men and women in your community

b) It would be very surprising to obtain the observed sample results if there is really no difference between men and women.

c) It would be very surprising to obtain the observed sample results if there is really a difference between men and women.

d) The probability is very small that there is no difference between men and women in your community on this issue.

e) The probability is very small that there is a difference between men and women in your community on this issue.


3) Suppose that the difference between the sample groups turns out not to be significant, even though your review of the research suggested that there really is a difference between men and women. Which conclusion is most reasonable?

a) Something went wrong with the analysis.

b) There must not be a difference after all.

c) The sample size might have been too small to detect a difference even if there is one.

4) Suppose that two different studies are conducted on this issue. Study A finds that 40 of 100 women sampled dream in color, compared to

20 of 100 men. Study B finds that 35 of 100 women dream in color, compared to 25 of

100 men. Which study (A or B) provides stronger evidence that there is a difference

between men and women on this issue?

a) Study A

b) Study B

c) The strength of evidence would be similar for these two studies.


5) Suppose that two more studies are conducted on this issue. Both studies find that 30% of women sampled dream in color, compared to 20% of men. But Study C consists of 100 people of each sex, whereas Study D consists of 40 people of each gender. Which study provides stronger evidence that there is a difference between men and women on this issue?

a) Study C

b) Study D

c) The strength of evidence would be similar for these two studies.

6) If the difference in the proportions (who dream in color) between the two samples does turn out to be statistically significant, which of the following is a possible explanation for this result?

d) Men and women in your community do not differ on this issue but there is a small probability that random chance alone led to the difference we observed between the two samples.

e) Men and women in your community differ on this issue.

f) Either (a) or (b) are possible explanations for this result.


7) Reconsider the previous question. Now think not about possible explanations but plausible (believable) explanations. If the difference in the proportions (who dream in color) between the two samples does turn out to be statistically significant, which of the following is the more plausible explanation for this result?

a) Men and women in your community do not differ on this issue but there is a small chance that random sampling alone led to the difference we observed between the two groups.

b) Men and women in your community differ on this issue.

c) They are both equally plausible explanations for this result.


Technology Desired features

Transparency: lets student “visualize” what the simulation/randomization is doing

Easy to use: Aids student learning, and does not become an obstacle in the path to learning statistics. Supervision useful, but not imperative.

Easily available: Can be run on different platforms, and is affordable

Provides choices to the user: For e.g. lets user pick the direction to look at to calculate p-value.

Interactive: With pop-up message asking, “Are you sure that’s the direction?” or “Entry does not match table”


Technology as a teaching tool How much of the thinking do we want the students to do

when they are using technology?


Example:

vs.


Example:


That’s all folks!


Requisite Knowledge for Teachers, Assessment Questions, Technology Allan Rossman, Soma Roy, Beth...

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