SOCR EduMaterials Activities BMI Modeling Activity

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(Exploratory data analyses (EDA))
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<center>[[Image:SOCR_Activity_BMI_ChiSquare_Fig2.png|700px]]</center>
<center>[[Image:SOCR_Activity_BMI_ChiSquare_Fig2.png|700px]]</center>
-
==Conclusions==
+
==Quantitative Data Analyses (QDA)==
 +
In this section, we will be testing the BMI variable for normality, although the same analysis can be carried for the other variables. As the name '''goodness-of-fit''' implies, we first need to create a normal model to compare to. We will use the sample mean (25.18643319) and standard deviation (3.146481308) as the parameters of the normal distribution.
 +
 
 +
The next few steps will use the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions Applet] (see [[SOCR_EduMaterials_DistributionsActivities|Distribution Activities]]). Open the [http://socr.ucla.edu/htmls/SOCR_Distributions.html applet] in a java-enabled browser.
 +
 
 +
<center>[[Image:SOCR_Activity_BMI_ChiSquare_Fig3.png|500px]]</center>
 +
 
 +
Enter in the values for the mean and standard deviation, then drag the graph down to see your full distribution.
 +
 
 +
<center>[[Image:SOCR_Activity_BMI_ChiSquare_Fig4.png|500px]]</center>
 +
 
 +
To run a goodness of fit test, we will need to create a set of bins to compare between the real distribution and the '''expected''' normal one. For simplicity’s sake, we will use a 16 bins of bin-size 1 beginning with BMI=18 and ending with BMI=34. To find the frequency of results in each bin from the normal distribution, click on the edges of the bin size (try to be as accurate as possible) on the normal distribution applet. The example shown alternatively, use the [[AP_Statistics_Curriculum_2007_Normal_Std|normal CDF function]].
 +
 
 +
<center>[[Image:SOCR_Activity_BMI_ChiSquare_Fig5.png|500px]]</center>
 +
 
 +
After calculating the probability of each bin, multiply each of these probabilities by the total number of cases (in this case, 248). Now we can place these calculated frequencies next to the frequencies from the observed distribution (the observed frequencies were found by plain counting):
 +
 
 +
<center>
 +
{| class="wikitable" style="text-align:center; width:30%" border="1"
 +
|-
 +
!Bin(Simplified)||Bin(Actual)||Normal_Probability||EstimatedNormalFrequency||ObservedFrequency
 +
|-
 +
|18-19||18.000-18.999||0.013454035||3.39041682||1
 +
|-
 +
|19-20||19.000-19.999||0.025001757||6.300442764||6
 +
|-
 +
|20-21||20.000-20.999||0.042032155||10.59210306||11
 +
|-
 +
|21-22||21.000-21.999||0.06392769||16.10977788||23
 +
|-
 +
|22-23||22.000-22.999||0.087962245||22.16648574||20
 +
|-
 +
|23-24||23.000-23.999||0.109497598||27.5933947||38
 +
|-
 +
|24-25||24.000-24.999||0.123313845||31.07508894||26
 +
|-
 +
|25-26||25.000-25.999||0.125638222||31.66083194||31
 +
|-
 +
|26-27||26.000-26.999||0.115806485||29.18323422||26
 +
|-
 +
|27-28||27.000-27.999||0.096570457||24.33575516||21
 +
|-
 +
|28-29||28.000-28.999||0.072854419||18.35931359||9
 +
|-
 +
|29-30||29.000-29.999||0.049724263||12.53051428||15
 +
|-
 +
|30-31||30.000-30.999||0.030702836||7.737114672||10
 +
|-
 +
|31-32||31.000-31.999||0.017150734||4.321984968||6
 +
|-
 +
|32-33||32.000-32.999||0.008667207||2.184136164||2
 +
|-
 +
|33-34||33.000-33.999||0.00396249||0.99854748||3
 +
|}
 +
</center>

Revision as of 22:24, 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

Statistic 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)

Before we run any quantitative tests, let’s examine what these variables look like in graphical form. Keep an eye out for which variables appear to follow a normal distribution.

Quantitative Data Analyses (QDA)

In this section, we will be testing the BMI variable for normality, although the same analysis can be carried for the other variables. As the name goodness-of-fit implies, we first need to create a normal model to compare to. We will use the sample mean (25.18643319) and standard deviation (3.146481308) as the parameters of the normal distribution.

The next few steps will use the SOCR Distributions Applet (see Distribution Activities). Open the applet in a java-enabled browser.

Enter in the values for the mean and standard deviation, then drag the graph down to see your full distribution.

To run a goodness of fit test, we will need to create a set of bins to compare between the real distribution and the expected normal one. For simplicity’s sake, we will use a 16 bins of bin-size 1 beginning with BMI=18 and ending with BMI=34. To find the frequency of results in each bin from the normal distribution, click on the edges of the bin size (try to be as accurate as possible) on the normal distribution applet. The example shown alternatively, use the normal CDF function.

After calculating the probability of each bin, multiply each of these probabilities by the total number of cases (in this case, 248). Now we can place these calculated frequencies next to the frequencies from the observed distribution (the observed frequencies were found by plain counting):

Bin(Simplified)Bin(Actual)Normal_ProbabilityEstimatedNormalFrequencyObservedFrequency
18-1918.000-18.9990.0134540353.390416821
19-2019.000-19.9990.0250017576.3004427646
20-2120.000-20.9990.04203215510.5921030611
21-2221.000-21.9990.0639276916.1097778823
22-2322.000-22.9990.08796224522.1664857420
23-2423.000-23.9990.10949759827.593394738
24-2524.000-24.9990.12331384531.0750889426
25-2625.000-25.9990.12563822231.6608319431
26-2726.000-26.9990.11580648529.1832342226
27-2827.000-27.9990.09657045724.3357551621
28-2928.000-28.9990.07285441918.359313599
29-3029.000-29.9990.04972426312.5305142815
30-3130.000-30.9990.0307028367.73711467210
31-3231.000-31.9990.0171507344.3219849686
32-3332.000-32.9990.0086672072.1841361642
33-3433.000-33.9990.003962490.998547483


Practice problems

See also

References



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