SOCR EduMaterials Activities BMI Modeling Activity

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! colspan=2|Underwater_Density_(\( \fract{g}{cm^3}\)) || Body_Fat || Height(m) || Weight_(kg) ||  BMI
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! colspan=2|Underwater_Density_(\( \frac{g}{cm^3}\)) || Body_Fat || Height(m) || Weight_(kg) ||  BMI
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| Mean || 1.0562 || 18.854 || 1.787 || 80.547 || 25.18643319
| Mean || 1.0562 || 18.854 || 1.787 || 80.547 || 25.18643319

Revision as of 22:07, 30 January 2013

Contents

SOCR Educational Materials - Activities - SOCR Body Mass Index (BMI) Activity and Applications of the Chi-Squared Test

Often times when solving a problem from intro-level textbooks, we are told to assume that a population follows a normal distribution. Other times, a graph of the data will allow us to assume some degree of normality. This allows the use of a number of statistical analyses later on.

Motivation and Goals

The following activity will demonstrate one of the ways to test for normality, using the Chi-Squared test for Goodness-of-Fit. The model to fit will be the normal model. We will run this test on a human characteristic often assumed to fit at least some kind of normal model: BMI.

Summary

This activity uses a simplified version of the BMI data sets found here. Four cases of data were excluded due to extremely high BMIs that hinted at a mistake in the entry process. 10 variables from the original dataset were left out in the dataset presented here, though the same process presented here may be used on them for additional practice.


Data

Data Description

  • Number of cases: 248
  • Variables
    • Underwater Density – Density determined via a graduated-cylinder type test
    • Body fat—Calculated body density and tissue-type proportions using Siri’s equation (see the full dataset page)
    • Height
    • Weight
    • BMI—Body Mass Index, calculated as \( \frac{weight}{height^2} \).

Data Summary

Underwater_Density_(\( \frac{g}{cm^3}\)) Body_Fat Height(m) Weight_(kg) BMI
Mean 1.0562 18.854 1.787 80.547 25.18643319
SD 0.0184 8.0663 0.0659 12.0076 3.146481308

Raw Dataset

Underwater_Density(g/cm3)Body_FatHeight(m)Weight(kg)BMI
1.070812.31.7208569.9666223.6268
1.08536.11.8351578.5848823.33436
1.041425.31.6827569.8532224.66876
1.075110.41.8351583.8011924.88325
1.03428.71.8097583.5743925.51738
1.050220.91.8986595.367826.45525
1.054919.21.7716582.1002226.15703
1.070412.41.841579.8322623.54155
1.094.11.879686.6361424.5227
1.072211.71.866989.9246925.80102
1.0837.11.892384.4815823.59294
1.08127.81.930497.9759526.29208
1.051320.81.765381.8734226.27277
1.050521.21.8097593.0998328.42574
1.048422.11.765385.1619727.32805
1.051220.91.676473.8221626.26827
1.0333291.803488.7907127.3013
1.046822.91.803494.914229.18415
1.0622161.7208583.347628.14538
1.06116.51.866996.0481827.55796
1.055119.11.727281.1930327.21658
1.06415.21.7716590.9452728.97505
1.063115.61.7335563.6163321.16878
1.058417.71.77867.4718721.34319
1.0668141.7208568.6058523.16728
1.09113.71.816172.2345821.90109
1.08117.91.714559.647420.29161
1.046822.91.714567.1316722.83771
1.0913.71.6446560.4411822.34529
1.0798.81.752672.9149723.73838
1.071611.91.8732582.5538123.52587
1.08625.71.8097572.6881822.19354
1.071911.81.8097576.2035223.26686
1.050221.31.803499.1099330.47425
1.026332.31.8669112.150732.17806
1.010140.11.65186.9763431.90854
1.043824.21.77891.7390629.01956
1.034628.41.7335589.244329.69667
1.025832.61.701892.0792531.79397
1.027931.61.77898.4295431.13594
1.0269321.816196.1615829.15561
1.08147.71.727256.8124419.044
1.06713.91.8605574.5025521.52229
1.074210.81.714560.5545820.60023
1.06655.61.8097567.3584720.56625
1.067813.61.739961.5751620.34028
1.090341.6954557.8330320.11898
1.075610.21.8351571.7809921.31407
1.0846.61.752663.1627420.56342
1.080781.7208562.2555521.02287
1.08486.31.866969.2862319.87947
1.09063.91.714561.8019621.02458
1.047322.61.828889.8112926.85335
1.052420.41.727282.3270227.5967
1.0356281.765391.2854629.29305
1.02831.51.7970591.8524528.44267
1.04324.61.6700581.5332329.23316
1.039626.11.8605597.9759528.30328
1.031729.81.739981.0796426.78325
1.029830.71.7843587.6567327.5312
1.040325.81.701880.7394427.87845
1.026432.31.77893.2132329.48588
1.0313301.714583.234228.31567
1.049921.51.7970568.7192421.27933
1.067313.81.816170.1934221.28222
1.08476.31.7589570.4202222.76095
1.069312.91.816171.100621.55727
1.043924.31.816175.9767223.03568
1.07888.81.7462566.5646821.82886
1.07968.51.8732572.9149720.77903
1.06813.51.625656.6990521.45598
1.07211.81.6700564.8637123.25642
1.066618.51.714567.2450722.87628
1.0798.81.765373.7087623.65277
1.048322.21.739980.6260426.63341
1.049821.51.7843573.1417722.97235
1.05618.81.7589577.6776925.10668
1.028331.41.7208574.2757525.08193
1.038226.81.7081568.1522523.3576
1.056818.41.8478586.2959525.27301
1.0377271.77877.450924.49982
1.0378271.7589576.2035224.63021
1.038626.61.714575.7499325.76958
1.064814.91.7081571.554224.52354
1.046223.11.6700572.5747826.02117
1.088.31.841580.1724523.64186
1.066614.11.854279.8322623.22016
1.05220.51.77880.2858525.3966
1.057318.21.765381.5332326.16361
1.07958.51.790774.9561423.37553
1.042424.91.8224587.3165326.28968
1.078591.892383.5743923.33959
1.099117.41.97485101.831526.11042
1.0779.61.8605585.6155624.73261
1.07311.31.689173.7087625.83499
1.058217.81.7335570.9872123.62149
1.048422.21.828889.357726.71773
1.050621.21.866990.0380925.83355
1.052420.41.828878.8116723.56449
1.05320.11.8097578.3580823.92471
1.04822.31.8732589.244325.4325
1.041225.41.7589580.2858525.94968
1.0578181.739975.0695424.79792
1.054719.31.866990.8318726.0613
1.056918.31.8859592.1926525.92006
1.059317.31.917787.9969223.92799
1.0521.41.7589576.4303124.70351
1.053819.71.739977.450925.58456
1.0355281.77883.120826.29337
1.048622.11.77880.8528425.57595
1.050321.31.7843573.9355623.22166
1.038426.71.8224579.4920623.93385
1.060716.71.7589571.6675923.16412
1.052920.11.8478580.3992523.54608
1.067113.91.828881.1930324.27651
1.040425.81.879686.6361424.5227
1.057518.11.8351585.0485725.25363
1.035827.91.892393.6668226.15808
1.041425.31.816184.0279925.47678
1.065214.71.7462572.6881823.83696
1.0623161.6954568.7192423.90608
1.067413.81.689173.0283725.59652
1.058717.51.701875.7499326.15563
1.037327.21.7462580.5126526.40288
1.05917.41.7208569.0594423.32046
1.051520.81.8605587.2031325.19123
1.064814.91.7716574.9561423.88094
1.057518.11.816177.9044923.62017
1.047222.71.790777.6776924.22427
1.045223.61.8605589.357725.81364
1.039826.11.6954571.21424.77396
1.043524.41.765376.3169224.48972
1.037427.11.7716584.3681826.8796
1.049121.81.7970575.6365323.42131
1.032529.41.879685.1619724.10543
1.048122.41.8097576.3169223.30149
1.052220.41.90596.5017826.59165
1.042224.91.803480.1724524.65137
1.057118.31.765378.5848825.2175
1.045923.31.7208575.7499325.57974
1.07759.41.8351572.4613821.5161
1.075410.31.968585.343422.02415
1.066414.21.7970570.7604121.91139
1.05519.21.8478594.5740127.69736
1.032229.61.7716593.6668229.84214
1.08735.31.841565.203919.22782
1.041625.21.78435101.151131.76951
1.07769.41.752669.0594422.48316
1.054219.61.8923109.65630.62333
1.075810.11.8351566.2244919.66416
1.06116.51.7081571.100624.36808
1.051211.866990.8318726.0613
1.059417.31.9113577.7910921.29362
1.028731.21.752693.3266330.38365
1.0761101.8351582.7806124.5802
1.070412.51.7462561.9153620.30419
1.047722.51.816180.3992524.37656
1.07759.41.8351568.6058520.37127
1.065314.61.854288.904125.85882
1.069131.7462583.5743927.40693
1.064415.11.790763.5029319.80378
1.03727.31.828899.2233329.66753
1.054919.21.8732598.4295428.05007
1.049221.81.727275.4097325.27797
1.052520.31.83515101.944930.27069
1.01834.31.7653103.532533.22306
1.06116.51.765378.3580825.14472
1.092631.7208569.0594423.32046
1.09830.71.663757.0392420.60742
1.052120.51.803480.3992524.7211
1.060316.91.816179.9456624.23904
1.041425.31.82245102.852130.9672
1.07639.91.7589565.8842921.29486
1.068913.11.701868.4924523.6497
1.031629.91.8161109.429233.17827
1.047722.51.7589584.9351727.45242
1.060316.91.8923106.480829.7366
1.038726.61.8859599.4501327.9605
1.108901.727253.750718.01768
1.072511.51.7081566.1110922.65804
1.071312.11.7716572.2345823.01385
1.058717.51.8859577.337521.74352
1.07948.61.816175.9767223.03568
1.045323.61.88595105.573629.68212
1.052420.41.828895.4811928.54864
1.05220.51.841591.7390627.05271
1.043424.41.7335583.9145927.92317
1.072811.41.7589569.3996322.43108
1.01438.11.9304110.789929.73073
1.062415.91.790787.7701227.37165
1.042924.71.89865101.944928.27976
1.04722.81.8478573.8221621.61988
1.041125.51.7335581.6466327.16849
1.0488221.752670.8738123.07386
1.058317.71.816176.2035223.10444
1.08416.61.8478575.8633222.21767
1.046223.61.714577.450926.34823
1.070912.21.7843580.8528425.39424
1.048422.11.7589568.0388621.99126
1.03428.71.816190.9452727.57405
1.085461.879683.46123.62396
1.020934.81.77165101.151132.22662
1.06116.61.854294.6874127.54096
1.02532.91.663775.2963327.20344
1.025432.81.841588.4505126.08296
1.07719.61.7843572.8015822.8655
1.074210.81.7970572.4613822.43811
1.08297.11.727263.7297321.36273
1.037327.21.892398.0893527.39314
1.054319.51.8224576.3169222.97786
1.056118.71.7970588.3371127.35413
1.054319.51.854278.3580822.79138
1.067813.61.7716567.6986621.56871
1.08197.51.77870.0800222.16821
1.043324.51.8224590.3782827.21152
1.0646151.7589570.0800222.65099
1.070612.41.790769.5130321.67807
1.0399261.83515104.326230.97778
1.072611.51.714573.3685724.95945
1.08745.21.7081564.5235122.11393
1.07410.91.7462581.5332326.73756
1.070312.51.6954557.3794319.96118
1.06514.81.7335576.8839125.58366
1.041825.21.8859590.0380925.3143
1.064714.91.765379.1518725.39944
1.0601171.739976.0901225.13505
1.074510.61.6700567.0182724.02892
1.06216.11.8224582.6672124.88984
1.063615.41.816179.6054624.13589
1.038426.71.7081573.3685725.14537
1.040325.81.714571.554224.34222
1.056318.61.714576.5437126.03961
1.042424.81.8351586.8629425.79238
1.037227.31.765399.4047731.89849
1.070512.41.765370.4202222.5975
1.031629.91.6700586.0691530.85948
1.0599171.6700557.8330320.73562
1.0207351.73355101.831533.88515
1.030430.41.8288106.25431.76968
1.025632.61.84785103.305730.25456
1.0334291.739990.4916829.89235
1.064115.21.7589570.5336122.7976
1.030830.21.790797.7491630.48368
1.0736111.701860.8947821.02631
1.023633.61.7716591.1720729.04731
1.032829.31.676484.7083830.14193
1.0399261.790786.5227426.98265
1.027131.91.77894.1204229.77285

Exploratory data analyses (EDA)

Various data patterns may be observed and explored using different types of graphical tools for plotting variables. Which of the following graphs are more or less likely to demonstrate visually significant grouping differences?


Conclusions

Practice problems

See also

References



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