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IB Math Studies Internal Assessment:
Shoe Size and Height
School Name: International School of BangkokDate: November 2010Course: IB Math Studies
Statement and Plan of Task:
In this assessment I will investigate the relationship between shoe size and height. For this topic I have collected data from students in my age group, which is 17 to 18 years old. I collected fifteen shoe sizes and height for each gender. My task to is to find patterns, which reveal how they are correlated, and the chi-squared test will prove how significant the correlation is. In the end I will compare female and male results to see how the correlation results differ or if they are exactly the same.
Hypothesis:
I believe that the relationship between shoe size and height is significant. The larger shoe size is the taller a person will be.
The Measurements:
I have converted all the height measurements to centimeters and collected all shoe sizes in by American standards.
DATA COLLECTION
MALEHeight (X)Shoe Size (Y)XYXY
173929929811557
1689.52822490.251596
19010.536100110.251995
17711.531329132.252035.5
18010324001001800
1'6810.528224110.251764
17710313291001770
17212259841442064
18211331241212002
18012324001442160
18511.534225132.252127.5
18411338561212024
17210259841001720
16510272251001650
17810.531684110.251869
4620171696.528134
FEMALE
Height (X)Shoe Size (Y)XYXY
170728900491190
172825984641376
1677.52788956.251252.5
160725600491120
160725600491120
1777.53132956.251327.5
155924025811395
170728900491190
175930625811575
160725600491120
170928900811530
170828900641360
14652131625730
15562402536930
167927889811503
409082870.518719
MAXIMUM, AVERAGE, MINUMUN & MODE
Male
Height (cm)Shoe Size
Maximum190 12
Average17710.6
Minimum1659
Mode17710
Female Height (cm)Shoe Size
Maximum1759
Average1657.5
Minimum1465
Mode1707
BothHeight (cm)Shoe Size
Maximum19012
Average1719.05
Minimum1465
Mode1709
CORRELATION & LINEAR FIT
MALE GRAPH
WORKING OF CORRELATION
SxSy
=
=
=
= 6.9
=
=
=
= 0.86
= = 2.262
Solving for R:
= = 0.38
The wokring of the correlation matches up with the results of the graph.
FEMALE GRAPH
WORKING OF CORRELATION
SxSy
=
=
=
= 8.314
=
=
=
= 1.1323
= = 10.43
Solving for R:
= = 0.33
BOTH GENDER GRAPH
SxSy
=
=
=
= 170.97
=
=
=
= 9.446
= = 1575.983
Solving for R:
= = 0.577
CHI SQUARED TEST
MALE
Null Hypothesis- Height and shoe size are independent of each other.Alternative Hypothesis- Shoe size is dependent of height.
Observed Frequency ():
Height (cm)9-9.510-10.511-11.512-12.5Total
161-17012003
171- 18014128
181-19001304
Total274215
Expected FREQUENCY ():
Height (cm)9-9.510-10.511-11.512-12.5Total
161-170(2 x 3)/15= .4(7 x 3)/15=1.4(4 x 3)/15=.8(2 x 3)/15= .43
171-180(2 x 8)/15= 1.067(7 x 8)/15=3.733(4 x 8)/15=2.133(2 x 8)/15= 1.0678
181-190(2 x 4)/15= .533(7 x 4)/15=1.867(4 x 4)/15=1.066(2 x 4)/15= .5334
Total 274215
Calculated Chi Squared
-(-)(-)/
1.4.60.360.9
21.4.60.360.9
0.8-.80.640.8
0.4-.40.160.4
11.0670.0670.00450.0042
43.733.2670.07130.0191
12.133-1.1331.2840.602
21.067.9330.8700.815
0.533-.5330.2840.533
11.867-.8670.7520.4027
31.0671.9333.7363.501
0.533-0.5330.2840.533
9.41
Degree of Freedom = (row -1) x (column -1) = (3-1) x (4-1)= 6
With the significant level of 5% Chi Squared from the table equals to 12.59. The calculated result is 9.41 while the table result is 12.59. This means that the null hypothesis is accepted, meaning that height and shoe size are independent of each other.
FEMALE:
Null Hypothesis- Height and shoe size are independent of each other.Alternative Hypothesis- Shoe size is dependent of height.
Observed Frequency ():
Height (cm)5-5.56-6.57-7.58-8.59-9.5Total
141-150100012
151-160013004
161-170003126
171-180001113
Total1172415
Expected FREQUENCY ():
Height (cm)5-5.56-6.57-7.58-8.59-9.5Total
141-1500.1330.1330.9330.2670.5332
151-1600.2670.2671.8670.5331.0674
161-1700.40.42.80.81.66
171-1800.20.21.40.40.83
Total1172415
Calculated Chi Squared
-(-)(-)/
10.1330.8670.7525.65
00.133-0.1330.0170.127
00.933-0.9330.870.932
00.267-0.2670.0710.265
10.5330.4670.2180.409
00.267-0.2670.0710.265
10.2670.7330.5372.01
31.8671.1331.2830.687
00.533-0.5330.2840.532
01.067-1.0670.2840.266
00.4-0.41.1381.067
00.4-0.5330.160.4
32.80.20.2840.101
10.80.20.040.05
21.60.40.160.1
00.2-0.20.040.2
00.2-0.20.040.2
11.4-0.40.160.114
10.40.60.360.9
10.80.20.040.05
14.325
Degree of Freedom = (4-1) x (5-1)= 12
With the significant level of 5% Chi Squared from the table equals to 21.0. The calculated result is 14.325 while the table result is 21.0. This means that the null hypothesis is accepted, meaning that height and shoe size are independent of each other.
BOTH:
Null Hypothesis- Height and shoe size are independent of each other.Alternative Hypothesis- Shoe size is dependent of height.
Observed Frequency ():
Height (cm)5-6.57-8.59-10.511-12.5Total
141-15010102
151-16013004
161-17004509
171-180026311
181-19000134
Total2913630
Expected FREQUENCY ():
Height (cm)5-6.57-8.59-10.511-12.5Total
141-1500.1330.60.8670.42
151-1600.2671.21.7330.84
161-1700.62.73.91.89
171-1800.7333.34.7672.211
181-1900.2671.21.7330.84
Total2913630
Calculated Chi Squared
-(-)(-)/
10.1330.8670.7525.65
00.6-0.60.360.6
10.8670.1330.0180.021
00.4-0.40.160.4
10.2670.7330.5372.011
31.21.83.242.7
01.733-1.7333.0031.733
00.8-0.80.640.8
00.6-0.60.360.6
42.71.31.690.626
53.91.11.210.31
01.8-1.83.241.8
00.733-0.7330.5370.733
23.3-1.31.690.512
64.7671.2331.520.319
32.20.80.640.291
00.267-0.2670.0710.267
01.2-1.21.441.2
11.733-0.7330.5370.31
30.82.24.846.05
26.933
Degree of Freedom = (5-1) x (4-1)= 12
With the significant level of 5% Chi Squared from the table equals to 21.0. The calculated result is 26.933 while the table result is 21.0. Because this result is greater than the chi squared from the table, the null hypothesis is rejected. Which means that height and shoe size are dependent of each other.
Conclusion
After analyzing the gathered date and finding the R values and Chi squared test it can be concluded that height and shoe size are correlated. It is not a very strong correlation but with results of 0.38 for male, 0.33 for female and 0.577 for both the correlation is shown. In the graphs it is also very visible to see the correlation and because the point are not extremely spread out the it is shown that the correlation is somewhat strong. Chi squared showed us that the null hypothesis was accepted for both male and female but when it came to testing both, the null hypothesis was rejected.
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