Some Lessons Learned in Developing anIntroductory Online Undergraduate

Statistics CourseStatistics Course

Dr. Eugene L. Round

Improving Statistics Teaching

GAISE Guidelines

Randomization and bootstrapping techniques

Student projects in an introductory statisticsclass

Developing a nine-week introductoryDeveloping a nine-week introductoryundergraduate statistics course thatincorporates these ideas

AND MOST IMPORTANT, your input ontechniques used at other schools as well ascomments on our process will be sought in aneffort to continue to improve course delivery.

GAISE - Guidelines for Assessment andInstruction in Statistics Education

1. Emphasize statistical literacy and developstatistical thinking.

2. Use real data

3. Stress conceptual understanding rather thanmere knowledge of procedures.mere knowledge of procedures.

4. Foster active learning in the classroom

5. Use technology for developing concepts andanalyzing data.

6. Use assessment to improve and evaluatestudent learning.

1. Emphasize statistical literacyand develop statistical thinking.

Use technology and show students how to usetechnology to:

● Manage and explore data

● Perform inference

● Check conditions that underlie inference● Check conditions that underlie inferenceprocedures.

Include open-ended problems and projects.

Give students practice in choosing techniques.

2. Use real data

3. Stress conceptual understanding ratherthan mere knowledge of procedures.

Primary goal – Discover Concepts, not CoverMethods

Pare down content to focus on core conceptsPare down content to focus on core concepts

Perform routine computations using technology toallow greater emphasis on interpretation ofresults.

4. Foster active learning in theclassroom

Precede computer simulations withphysical explorations.

Don’t use activities that lead studentsstep-by-step through a list of procedures.

Include assessment as an importantInclude assessment as an importantcomponent of an activity.

5. Use technology for developingconcepts and analyzing data.

Implement computer intensive methods.

Allow students to focus on interpretation ofresults and testing of conditions rather than oncomputations.

Help students visualize concepts.Help students visualize concepts.

Explore “What if” questions.

HOWEVER, don’t replace a set of rote “by hand”procedures with a set of rote computerprocedures.

6. Use assessment to improve andevaluate student learning.

Students will value what you assess, soassessment should be aligned withlearning goals.

Use projects and open-ended investigativetasks.tasks.

Goals for our course

1. Integrate GAISE guidelines into a nine-weekcourse – classroom and online.

2. Reduce the number of techniques covered toconcentrate on understanding.

3. Use Homework and Testing software that wouldgive students immediate feedback.

4. Use Technology to explore data, reducecomputations, and give students time to explainresults.

5. Include a project.

6. Use computer intensive methods to get atconcepts

1. Reduce Coverage

Emphasize concepts that students traditionally havetrouble with

● Distributions, confidence interval, hypothesis testing,p-value

Cover only the Basics of Simple Linear Regression –Strength of RelationshipsStrength of Relationships

● Scatter plots, Correlation, Simple Linear RegressionLine

Cover only the Basics of Probability

● Randomness, Independence, Conditional Probability,Normal Probability Model

1. Reduce Coverage

Included (p-value method):

● One sample test for proportion

● Two sample test for proportion

● One sample t-test for mean

● Two sample t-test for mean paired data

● Two-sample t-test for mean, independent samples, w/o● Two-sample t-test for mean, independent samples, w/oequal variation assumption

Not Included:

● Two-sample t-test for mean assuming equal variation

● ANOVA

● Chi-Square

● Linear Regression

Technology

2. Use Homework and Testingsoftware that gives immediatefeedback - MyStatLab

3. Use Technology - StatCrunch3. Use Technology - StatCrunch

4. Include a Project http://faculty.ung.edu/DJSpence/NSF/materials.html

Incorporated a t-test project

●Made large data sets available for studentsto use or they can find their own data.

●Detailed scoring rubric given to students●Detailed scoring rubric given to students

●Must turn in project prior to taking finalexam

Excellent article at:http://www.amstat.org/publications/jse/v21n3/bailey.pdf

5. Use Computer Intensive Methods

StatCrunch incorporates the capability todo Bootstrapping and Randomizationmethods

Bootstrapping Illustration

Randomization Illustration

Bootstrapping- Reaction Times

• Dropper holds a meter stick vertically withone hand near the top of the stick with 0-centimeter mark is at the bottom.

• Catcher positions his or her thumb and indexfinger about 5 cm apart on opposite sides offinger about 5 cm apart on opposite sides ofthe meter stick at the bottom.

• Dropper drops the meter stick withoutwarning, and Catcher catches it.

• Location of the middle of the thumb of thecatcher is recorded (reaction distance)

Bootstrapping

Assume the first student isright handed. Then thereaction distance for theright hand was 8.0centimeters and for the leftcentimeters and for the lefthand was 8.5 centimeters.

File shows the reactiondistances of all 40students.

Bootstrapping

In a bootstrap sample, we take theoriginal values from the sample of 40and select another sample of size 40 byselecting 40 values with replacementfrom the original sample.from the original sample.

Bootstrapping allows us to quickly takea large number of samples from ourpopulation and plot the distribution ofsample statistics, such as the samplemean.

Bootstrapping-In Class Activity

We can simulatethis resamplingtechnique with a“hands-on”example. Supposeexample. Supposewe have 40 cardswith the originaldominant handreaction distanceswritten on them asshown.

Bootstrapping-In Class Activity

We want to construct abootstrap sample ofsize 40. First we wouldshuffle the cards.

Then we would selectone card from theshuffled deck.

Bootstrapping-In Class Activity

We wouldrecord thatnumber as partof ourof ourbootstrapsample on thespreadsheet asshown.

Bootstrapping-In Class Activity

We would put that card back into the deck,shuffle again, and select another card.

Bootstrapping-In Class Activity

We selected13.5, so wewould recordwould recordit on ourspreadsheet.

Bootstrapping-In Class Activity

Continue the process until we had a newsample of size 40.

If each student in a classroom course with 25students went through this process, we wouldhave 25 bootstrap samples that we could usehave 25 bootstrap samples that we could usein our analysis. We could plot the samplemeans and look at the distribution, forexample.

Time consuming process! Automate usingStatCrunch

Bootstrapping with StatCrunch

StatCrunch Demonstration

●File Hands Data

●www.statcrunch.com

Bootstrapping with StatCrunch

o This display fromStatCrunch shows thedistribution of the 3006sample means.

o Students often havedifficulty understandingwhat a “distribution” is.

o The graphic is apicture of thedistribution of samplemeans. How would you describe

the shape and spread?

The screen shots below show the results from 3000resamples of student/faculty ratio data. They givethe 2.5th , 5th , 95th, and 97.5th percentiles. We canuse those to construct bootstrap confidenceintervals and discuss the meaning of a CI.

Another Bootstrap Example

Here is ahistogram forthe student-faculty ratiofaculty ratiodata. Howwould youdescribe theshape of thedistribution?

Another Bootstrap Example

What is theshape ofthedistributionof samplemeans fromthe 3000

Distribution ofSample Means

Original Distribution

the 3000bootstrapsamples?

Randomization

Use randomization techniquesto lay groundwork forhypothesis testing.

The data in the file representthe amounts spent on clothingduring a one month period forduring a one month period forrandom samples of 14 collegewomen and 19 college men.

Our hypotheses are:

H0: µ(women) = µ(men)

Ha: µ(women) ≠ µ(men)

Randomization

In conducting the hypothesis test, weassume that H0 is true.

A key idea in the randomization techniqueis that if the null is true and there is nodifference in the average monthly amountsdifference in the average monthly amountsspent on clothing by women and by men,then any of the data points could comefrom a man or a woman.

Randomization-In Class Activity

In a classroom course,we would firstsimulate this in ahands-on activityusing cards asusing cards asfollows. We have twosets of cards, oneshowing the data forwomen and anothershowing the data formen.

Randomization-In Class Activity

If any of the data values in the picture could comefrom either women or men, then we can put all thecards together in one pile, reshuffle them, and thenform new samples of 14 values for the women and19 values for the men.

Putting all the Reshuffling the Forming newPutting all thecards together

Reshuffling thecards

Forming newsamples

Randomization-In Class Activity

The new sets ofdata make up ournew randomizeddistribution. In aclass with 25class with 25students, we wouldhave 25“randomizations”and we could lookat the distributionsof sample means.

Randomization using StatCrunch

This a somewhat time-consumingprocess. Its purpose is to givestudents a feel for what is going on inrandomization.

We will turn to technology and useStatCrunch to do this now.

Use file Clothes Data

www.statcrunch.com

Randomization

In the display, there is aregion for -39.62406 orbelow and +39.62406 andabove.

As we do the simulation,we will be interested in thewe will be interested in thenumber and proportion ofresults that fall outside ofthose values. Those wouldbe the results that give“more extreme” values thanwhat we got in the originalsamples.

Randomization

Notice that on the left sideof the display most of thesamples are shown inblack, but some are in red.Those in red are the onesin which the difference inin which the difference insample means was moreextreme than what we hadin our original data, that is,less than -39.62406 orgreater than +39.62406.

RandomizationThe right side of the displayshows that 210 or 6.99 percentof the samples had a differencein sample means less thanwhat we got in our originalsample and 234 or 7.78 percenthad a difference in samplemeans greater than what wegot in our original sample. Agot in our original sample. Atotal of 444 or 14.77% of the3006 samples resulted in adifference in sample meansmore extreme than what we gotin the original data. Thoseresults are also shown in red inthe histogram.

Randomization

To summarize—the question we want to answer is whetheror not the results of our original sample show an “unusual”occurrence—something that would not happen just bychance but could be the result of a real difference inspending habits of college women and men.

We assumed there is no difference in spending and ran We assumed there is no difference in spending and ran3006 simulations based on that assumption.

Approximately 15% of the simulations resulted indifferences in mean spending more extreme than what wegot in our original sample. Stated another way, oursimulation showed that even if there is no difference inmean spending between the two genders, we will get adifference at least as great as what we got in our originaldata approximately 15% of the time.

Randomization

Do our results support the null hypothesis of nodifference in mean spending or the alternate hypothesisthat there is a real difference?

The answer depends on whether or not we think anevent that happens 15% of the time is unusual.

We would probably say that something that can happenWe would probably say that something that can happen15% of the time is not too unusual so there is notsufficient evidence to reject the null hypothesis.

We would conclude that there isn’t a significantdifference in mean monthly amounts spent on clothes.

Even though the original data showed a difference, thatdifference isn’t large enough for us to conclude there isa real difference.

Goals for our course

Integrate these guidelines into a nine-week course– classroom and online.

Reduce the number of techniques covered toconcentrate on understanding.

Use Homework and Testing software that wouldgive students immediate feedback.

Use Technology to explore data, reducecomputations, and give students time to explainresults.

Include a project.

Use computer intensive methods to get atconcepts

Some Lessons Learned in Developing anIntroductory Online Undergraduate

Statistics CourseStatistics Course

Dr. Eugene L. Round