Confident in a Kiss?

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Teaching Statistics. Volume 25, Number 1, Spring 2003

Blackwell Publishing LtdOxford, UKTESTTeaching Statistics0141-982X© Teaching Statistics Trust 200120022511000Original Article

Confident in a Kiss?

KEYWORDS:

Teaching; Active learning; Confidence interval for proportion; Simulation.

Mary Richardson

Grand Valley State University, Michigan, USA. e-mail: [email protected]

Susan Haller

Saint Cloud State University, Minnesota, USA. e-mail: [email protected]

Summary

This article describes an interactive activity forillustrating general properties of confidence inter-vals and the construction of confidence intervalsfor proportions. In completing this activity, studentsgenerate, collect and analyse data.

ª

INTRODUCTION

ª

I

n this article, we discuss an activity that wedeveloped for illustrating properties of confidence

intervals for proportions. The activity is completedinteractively in the classroom. The estimatedcompletion time for the activity is a one-hour classperiod.

We developed this activity to use in an introductorystatistical concepts course at the undergraduatelevel. The activity was designed to help studentsvisualize the process of repeatedly sampling froma population and using the sample data to makean inference about an unknown population para-meter. Through completion of this activity, studentsdevelop an understanding of how to interpret aconfidence interval and the relationship betweenthe confidence level and confidence intervalwidth.

Prior to completing this activity, students havestudied Normal distributions, basic probabilityconcepts, sampling distributions and basic elementsof confidence interval construction and interpreta-tion. Additionally, students have been introducedto the formula for an approximate (1

−

α

)100%confidence interval for the population propor-tion

p

(based on a random sample of size

n

,where

n

is large). The confidence interval for

p

isgiven by

where is an appropriate percentile of the standard

Normal distribution,

p

is the sample proportion,and both

n

p

and

n

(1

−

p

) are

≥

5.

Our intention in developing this activity was to helpstudents expand their knowledge about confidenceintervals beyond the simple mechanics of intervalconstruction. This activity engages the students andbrings life to a lecture on construction and inter-pretation of confidence intervals for a proportion.

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THE ACTIVITY

ª

Background

Students enjoy collecting and analysing data.Especially, it seems, when chocolate is involved. Inthis project, students explore confidence intervalsfor a proportion. Initially, the unknown proportionis the proportion of base landings for tossed plainHERSHEY’S

®

KISSES

®

chocolates. For readersnot familiar with HERSHEY’S

®

KISSES

®

, theymight best be described as small (~ 4 grams), cone-shaped chocolates individually wrapped in foil. Apicture of and information about these candiescan be found at the HERSHEY’S

®

Web site athttp://www.hersheys.com/products/kisses.shtml. A‘base landing’ for a candy occurs when it lands onthe circular face of the cone.

Materials

Students work in groups of three. Each group has30 plain KISSES

®

chocolates (10 for each student),pp p

( )

±−

zn

α2

1

zα2


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a 16-ounce plastic cup and a flat table or desktop onwhich to work. Each student has three sticky notes,a copy of the Activity Worksheet (see figure 1) anda copy of the Questions Sheet (see figure 2). (Thefigures are shown at the end of the article.)

The set up

To begin the activity, each student examines aKISSES

®

chocolate. (We tell the students theycan eat the candies later.) We note that there aretwo possible outcomes if a KISSES

®

chocolate istossed onto the desktop

−

landing on the base andlanding on the side. Each student is then askedto estimate the proportion of the time that aKISSES

®

chocolate will land on its base whentossed onto the desktop (estimates usually rangefrom 20% to 80%). After students make their esti-mates, they are ready to conduct the followingexperiment.

Data collection

Each group puts 10 KISSES

®

chocolates into theplastic cup, and the following tasks are assigned tothe group members:

• one group member spills the candies onto thetable five times, each time counting the numberof candies that land on their base

• a second group member helps the spiller countthe candies

• the third group member records the results onthe spiller’s Activity Worksheet.

After one student has spilled the cup five times, theassigned duties rotate among group members.

In table 1, we show an example of typical individualoutcomes for the tosses of the KISSES

®

chocolates.

Note: in past experiments, the proportion of thetime that a plain KISSES

®

chocolate lands on itsbase when tossed has consistently been near35%.

Data analysis: confidence intervals

After the groups have completed spilling theKISSES

®

chocolates, the students are asked tobegin answering the questions on the QuestionsSheet (figure 2).

As was noted earlier, previous experience withtossing the KISSES

®

chocolates has shown thatthe proportion of base landings for spilled candiesis approximately

p

= 0.35.

First, the students are each asked to construct a90% confidence interval for the proportion of baselandings using individual data collected from allfive of their tosses (a sample of size 50). An exam-ple of a typical individual 90% confidence interval(based on the data given in table 1) is

After constructing the confidence interval, eachstudent writes it on a sticky note and gives it to theinstructor. An example of a typical class set of90% confidence intervals is shown in table 2.

We use an overhead transparency sheet to graphthe confidence intervals constructed by the class(see figure 3, at the end of the article). Each student’sconfidence interval is sketched horizontally on theoverhead transparency (with a thick transparencymarker), leaving one blank horizontal line betweenintervals. The resulting overhead transparency willdisplay all of the confidence intervals constructedby the class.

Graphing the confidence intervals on the overheadallows students to visualize the result of drawingmany samples from the same population and cal-culating a 90% confidence interval from each sam-ple. After graphing the class confidence intervals,we discuss their meaning. For a typical class of 30

Table 1. Example individual results for tossing the KISSES®

chocolates

Toss numberNumber of candies landing on base

1 52 73 24 35 1Total 18

1850

1 645

1850

11850

500 36 0 11 0 25 0 47 .

( ) . . ( . , . )±

−= ± =

Table 2. Example 90% class confidence intervals

StudentProportion of candies landing on base

90% confidence interval

1 0.38 (0.27,0.49)2 0.34 (0.23,0.45)3 0.32 (0.21,0.43): : :28 0.22 (0.12,0.32)29 0.28 (0.18,0.38)30 0.36 (0.25,0.47)

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students, roughly 27 (90%) of the class confidenceintervals should cover the value of

p

= 0.35. Westress that if we claim that we are 90% confidentthat a proportion lies within the endpoints of aconfidence interval, we are saying that the end-points of the confidence interval were calculatedby a

method that gives correct results in 90% of allpossible samples

. We then ask students to write astatement explaining how a 90% level of confidenceshould be interpreted.

Students are then each asked to construct a 99%confidence interval for the proportion of baselandings using individual data collected from allfive of their tosses. Using the same procedure, theclass confidence intervals are graphed on theoverhead transparency. After graphing the classconfidence intervals, we discuss the results. For atypical class of 30 students, roughly 29 or 30(99%) of the class confidence intervals shouldcover the value of

p

= 0.35. We stress how toproperly interpret a 99% confidence level and weask students to write a statement explaining howa 99% level of confidence should be interpreted.We also ask students to write a statement explain-ing how increasing the confidence level from 90%to 99% changed the width of their confidenceintervals.

Finally, students are each asked to construct an80% confidence interval for the proportion of baselandings using individual data collected from allfive of their tosses. Using the same procedure, theclass confidence intervals are graphed on theoverhead transparency. After graphing the classconfidence intervals, we discuss the results. For atypical class of 30 students, roughly 24 (80%) ofthe class confidence intervals should cover thevalue of

p

= 0.35. We stress how to properly inter-pret an 80% confidence level and we ask studentsto write a statement explaining how an 80% levelof confidence should be interpreted. We also askstudents to write a statement explaining howdecreasing the confidence level from 99% to 80%changed the width of their confidence intervals. Wethen ask students to write an intuitive justificationfor the relationship between the width of theconfidence intervals and the level of confidence.

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CONCLUSION

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This activity provides a fun and interesting way toillustrate properties of confidence intervals.

Through completing this activity, students willsee a demonstration of the process of repeatedlyselecting samples from a population and con-structing confidence intervals for the populationproportion.

Plotting class results on the overhead allows for adiscussion of the meaning of the level of confidence.Students can visualize that, in repeated sampling,the confidence level represents the percentage of thetime that the

process

of constructing a confidenceinterval will result in an interval that successfullyencloses the true value of the population parameter.Changing the confidence level allows students tosee the relationship between level of confidenceand confidence interval width.

To paraphrase Moore (2000), after completing thisactivity, students will have an understanding of whatstatistical confidence does

not

say. That is, if weclaim that we are 90% confident that a proportionlies within the endpoints of a confidence interval,we are saying that the endpoints of the confidenceinterval were calculated by a

method that givescorrect results in 90% of all possible samples

. Wecannot say that the probability is 90% that the trueproportion falls within the endpoints of the confidenceinterval. No randomness remains after we drawone particular sample and construct from it oneparticular interval. The true proportion either is oris not between the confidence interval endpoints.

Acknowledgement

The HERSHEY’S and KISSES trademarks are usedwith permission of Hershey Foods Corporation.

References

Moore, D. (2000).

The Basic Practice of Statis-tics

(2nd edn). New York: W.H. Freeman.


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Fig 1. Activity worksheet

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Fig 2. Questions Sheet


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Fig 3. Sheet for overhead projector transparency

Confident in a Kiss?

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