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  • Level: GCSE
  • Subject: Maths
  • Word count: 4731

Hypothesis"The taller you are, the bigger is your foot length". I think my hypothesis is true because there is a positive correlation between the heights and foot length.

Extracts from this document...

Introduction

Designing and Planning

Introduction

I am investigating a hypothesis using the data from the Census at School Data Selector, to see whether it is proved correct.

Hypothesis

“The taller you are, the bigger is your foot length”.

I think my hypothesis is true because there is a positive correlation between the heights and foot length.

Plan of action-collecting data

I will collect the data I need from the Census at School Data Selector. The data will be useful because it will give me an opportunity to do a preliminary example test on the hypothesis which I am trying to prove. And I will do histograms to see if my data was reliable. Then I will use the data from a statsfile which is a selection of results ranged from ages 7 to 17 collected in 2002-2003, it is a secondary data. I will collect 50 samples of random sampling and a fairer 100 samples of stratified sampling to test the hypothesis. I will try to make sure that my results are reliable by using the stratified sample I’m going to take from the statsfile. To test its reliability, I will do a histogram. I will have quantitative data because all my samples are numerical. My quantitative data will be continuous. I will record the data I collected in Microsoft Excel.

Plan of action- Processing and representing data

...read more.

Middle

F

14

172

25

674

F

14

154

23

591

F

14

180

25

646

F

14

171

24

698

F

14

154

23

572

F

14

166

24

742

M

14

210

35

685

F

14

161

23

639

F

14

165

22

726

F

14

172

26

746

M

14

170

25

658

F

14

160

24.5

909

M

14

156

23

857

M

14

182

27

708

F

14

162

25

488

F

14

165

19

893

M

14

176

30

716

F

14

166

24

871

M

14

176

30

653

F

14

155

22

650

F

14

173

27

509

F

14

200

24

913

M

14

178

30

701

F

14

169

22

986

M

15

180

27

973

M

15

176

23

985

M

15

191

28

990

M

15

181

29

949

F

15

171

23

973

M

15

176

23

936

F

15

160

21

958

F

15

154

19

991

M

15

175

24

987

M

15

110

27

1033

F

16

166

21.5

1065

M

16

190

35

1048

F

16

166

23

1035

F

16

168

26

Cumulative frequency

Height

Number of people

Cumulative frequency

110≤h<120

1

1

120≤h<130

2

3

130≤h<140

0

3

140≤h<150

4

7

150≤h<160

29

36

160≤h<170

31

67

170≤h<180

25

92

180≤h<190

4

96

190≤h<200

2

98

200≤h<210

1

99

210≤h<220

1

100

image00.png

It shows the measure of spread on the middle 50% of the data (between the upper and the lower quartile)

The stratified sample has no skew, mode=median=mean, as the median- lower quartile= higher quartile- median:

160- 153= 167- 160

7=7

From the graph, we can see that my data is reliable as the interquartile range is 14 cm which is relatively small. This shows that most of my data is gathered around the mean which follows the normal pattern we expects.

Standard deviation

It is mainly used to compare to sets of data. It’s another measure of spread or dispersion about the mean in a more accurate way than the range or interquartile range.  It gives a more detailed picture of the way in which the data is dispersed about the mean as the centre of the distribution.

Standard deviation (s.d) = image01.png

Where  represents the sum

x represents an item of data

n represents the number of item of data

The symbol image09.pngis the mean whereimage12.png

Variance = the mean of the squares minus the square of the mean

Height (cm) mid interval frequency

x

f

fx

x2

fx2

110≤h<120

115

1

115

13225

13225

120≤h<130

125

2

250

15625

31250

130≤h<140

135

0

0

18225

0

140≤h<150

145

4

580

21025

84100

150≤h<160

155

29

4495

24025

696725

160≤h<170

165

31

5115

27225

843975

170≤h<180

175

25

4375

30625

765625

180≤h<190

185

4

740

34225

136900

190≤h<200

195

2

390

38025

76050

200≤h<210

205

1

205

42025

42025

210≤h<220

215

1

215

46225

46225

                                       ∑

100

16480

2736100

Mean, image13.png=16480 / 100=164.8

Variance, image14.png= 2736100 / 100 – 164.82= 201.96

Standard deviation, s = √201.96= 14.21 (2 d.p), so s.d. of the height is 8.6%

Any values that don’t lie within 2 s.d. away from the mean are outliers:

164.8± 14.21*2

So values <136.38cm or >193.22cm are outliers.

Foot length (cm) mid interval frequency

x

f

fx

x2

fx2

15≤f<20

17.5

5

87.5

306.25

1531.25

20≤f<25

22.5

55

1237.5

506.25

27843.75

25≤f<30

27.5

33

907.5

756.25

24956.25

30≤f<35

32.5

5

162.5

1056.25

5281.25

35≤f<40

37.5

2

75

1406.25

2812.5

                                      ∑

100

2470

62425

...read more.

Conclusion

-8.29

-3.253

-0.71262

-4.83756

8.71

0.747

0.74872

1.11087

-9.29

-1.253

-0.79858

-1.86335

16.71

0.747

1.43642

1.11087

7.71

-0.253

0.66276

-0.37624

-9.29

-1.253

-0.79858

-1.86335

2.71

-0.253

0.23296

-0.37624

46.71

10.747

4.01526

15.98196

-2.29

-1.253

-0.19685

-1.86335

1.71

-2.253

0.14699

-3.35046

8.71

1.747

0.74872

2.59798

6.71

0.747

0.5768

1.11087

-3.29

0.247

-0.28281

0.36732

-7.29

-1.253

-0.62666

-1.86335

18.71

2.747

1.60834

4.08509

-1.29

0.747

-0.11089

1.11087

1.71

-5.253

0.14699

-7.81178

12.71

5.747

1.09257

8.54641

2.71

-0.253

0.23296

-0.37624

12.71

5.747

1.09257

8.54641

-8.29

-2.253

-0.71262

-3.35046

9.71

2.747

0.83469

4.08509

36.71

-0.253

3.15565

-0.37624

14.71

5.747

1.26449

8.54641

5.71

-2.253

0.49084

-3.35046

16.71

2.747

1.43642

4.08509

12.71

-1.253

1.09257

-1.86335

27.71

3.747

2.38199

5.5722

17.71

4.747

1.52238

7.0593

7.71

-1.253

0.66276

-1.86335

12.71

-1.253

1.09257

-1.86335

-3.29

-3.253

-0.28281

-4.83756

-9.29

-5.253

-0.79858

-7.81178

11.71

-0.253

1.00661

-0.37624

-53.29

2.747

-4.58089

4.08509

2.71

-2.753

0.23296

-4.09401

26.71

10.747

2.29603

15.98196

2.71

-1.253

0.23296

-1.86335

4.71

1.747

0.40488

2.59798

image10.png

image11.png

Conclusion

My original hypothesis was “the taller you are, the bigger is your foot length” and it’s supported by the overall evidence.

From my preliminary work we established that there is a positive correlation between the height and foot length. This enabled me to carry the investigation further, using the data from stratified sampling which are both random and fair.

I found out the interquartile range and standard deviation to prove the reliability of my results. The correlation coefficient of my sample is 0.38, showing some positive correlation. I also went further and did regression lines where we can estimate values of variable given values of the other if we know one of them.  

All this areas I have covered proves my hypothesis to be correct. And this investigation could be developed further. I could investigate the difference in the relationship between height and foot length in different ages, such as 13, 14, 15, 16, 17and 18 years old, where I would expect the relationship to be stronger with older ages as the body proportions varies more with younger children.

Census at School data

Height (cm)

Foot length (cm)

119

20

129

26

130

24

140

21

140

22

140

22

140

24

142

21

142

24

143

21.5

144

22

145

22

145

23.5

146

21

146

25.5

148

21

149

21

149

23

149

24

149

25

149

27

150

20

150

22

151

20

151

24

153

20

153

21

153

22

153

24

154

22

154

22.5

154

24

154

26

155

17

155

21

155

23

155

23

155

27

156

20

156

24

156

25

157

20.5

157

23.5

157

24.5

158

22

158

22

158

23

158

24

158

24

159

26

159

27

160

16

160

24

161

23

161

23.5

161

24

161

26

162

22

162

22

162

25

162

25

162

27

163

20

163

24

163

24.2

163

25.5

163

26.5

164

19

164

20

164

22

164

24

164

25

165

24

165

25

166

24

166

24

167

23

167

26

168

20

168

24

168

24.9

169

26

169

28

170

24

170

24

170

24

170

26

172

30

174

24

174

28

174

32

175

23

175

24

176

26

176

26

177

22

184

27

185

33.7

186

29

194

28

...read more.

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