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Estimating the length of the line and the size of the angle

Extracts from this document...

Introduction

Aziz Elgindi

Guestimate

Mathematics Coursework

          My task is to write a hypothesis and to test how well people estimate and to design and carry out an investigation to test my hypothesis.

        My hypothesis is that people will estimate the length of the line better than the size of the angle because lines are used more commonly than angles are. My hypothesis is also that neither the boys nor the girls will estimate better than one another. I also predict that people in higher sets will estimate the angles and lines better than people in lower sets.

        I will need to carry out an investigation to test my hypothesis. The investigation which I will carry out will be a questionnaire. The questionnaire will be conducted with two lines and two angles which I will ask twenty people in year eleven to estimate the length of the two lines and the size of the two angles. I have chosen twenty people because it will give me a set of eighty results, which is enough to prove my hypothesis. I will do this using a stratified sample to find out how many people I will ask in each set. Once I have found out how many people I will ask in each set, I will randomly pick an even number of boys and girls by picking boys and girls’ names out of a hat. If the number of people that I will ask is an odd number e.g. five, then I will pick two boys from one hat and two girls from the other, then I will mix the names of the boys and girls in one hat and pick out the last one.

...read more.

Middle

8.3

4.2

43

67

Chris

4

M

10.3

7

45

55

Emma

4

F

7.5

4.5

38

50

Chezirva

5

M

10

5

50

70

Alhassan

5

M

7.9

4.9

50

80

Becky

5

F

6.9

4

35

70

        Now that I have collected all of this data, the next thing that I need to do is to organise the data into different tables to see if my hypothesis is true. I need to do this so that I can easily look at the data that I need to e.g. different list of boys and girls. I will also need to draw some graphs to shows this information in other ways and to see if my hypothesis is correct.

        Now that I have my results I need see if my first hypothesis is true. To do this I need to find out the average length of the line and the average size of the angle. I will compare them by using 1 millimetre to 1 degree. Below is a table showing the results I got and the cumulative frequency which I will use to add up the results. Then I will need to find the mean which is recorded on the bottom of the table. To work out the mean, I will need to find out the total of each column and then divide it by 20. The reason that I am using cumulative frequency is to find the total of each row. The formula for finding the mean is:

Total of the records obtained    

Number of records obtained    

For example to find the mean of Line 1:

165.4   = 8.27cm

  20

Line 1 (cm)

Cum. Freq of Line 1

Line 2 (cm)

Cum. Freq of Line 2

Angle 1 (º)

Cum. Freq of Angle 1

Angle 2 (º)

Cum. Freq of Angle 2

6.6

6.6

4.7

4.7

44

44

72

72

6

12.6

4

8.7

30

74

80

152

8

20.6

6

14.7

50

124

52

204

8

28.6

4.5

19.2

45

169

75

279

9.4

38

5

24.2

55

224

82

361

8.5

46.5

5.5

29.7

40

264

70

431

7.5

54

5

34.7

45

309

60

491

9

63

3

37.7

45

354

55

546

10

73

5

42.7

40

394

65

611

9

82

4.5

47.2

45

439

60

671

8.5

90.5

6

53.2

50

489

65

736

7

97.5

4

57.2

70

559

80

816

8

105.5

4

61.2

25

584

70

886

9

114.5

5

66.2

45

629

75

961

8.3

122.8

4.2

70.4

43

672

67

1028

10.3

133.1

7

77.4

45

717

55

1083

7.5

140.6

4.5

81.9

38

755

50

1133

10

150.6

5

86.9

50

805

70

1203

7.9

158.5

4.9

91.8

50

855

80

1283

6.9

165.4

4

95.8

35

890

70

1353

Mean of line 1 =

8.27cm

Mean of line 2 =

4.79cm

Mean of Angle 1 =

44.5º

Mean of Angle 2 =

67.65º

        Now that I have found out the mean of each of the two lines and two angles, I need to see if my hypothesis was correct. I will do this by comparing the results of the angles and the results of the lines. I will compare the results by using 1 millimetre to 1 degree.

Line one was 8.7 cm and the mean that I got was 8.27cm. That is a difference of 43mm.

Line two was 5.3cm and the mean that I got was 4.79cm. That is a difference of 51mm.

Angle 1 was 37º and the mean that I got was 44.5º. That is a 7.5º difference.

Angle 2 was 63º and the mean that I got was 67.65º. That is a 4.65º difference.

        Looking at the results, I can see that my hypothesis was incorrect because the size of angle was guessed better than the size of the line. This could be because the comparison of 1 millimetre to 1 degree is not a very good comparison. It could also be because I chose the mean. Maybe the results would have been different if I used the mode or median or standard deviation. The reason which I chose to use the mean is that it takes in account all of the results that I obtained.

        Now I will need to design a new table to easily compare the boys and the girls. Then I will need to find the mean to see if my hypothesis is true.

Names of Boys

Line 1

Cum Freq

Line 2

Cum Freq

Angle 1

Cum freq

Angle 2

Cum Freq

Chris

6.6

6.6

4.7

4.7

44

44

72

72

Darren

8

14.6

6

10.7

50

94

52

124

Joe

10

24.6

5

15.7

40

134

65

189

Ian

9

33.6

4.5

20.2

45

179

60

249

Dom

8.5

42.1

6

26.2

50

229

65

314

Aidan

8

50.1

4

30.2

25

254

70

384

Rick

8.3

58.4

4.2

34.4

23

277

67

451

Chris

10.3

68.7

7

41.4

45

322

55

506

Chezirva

10

78.7

5

46.4

50

372

70

576

Alhassan

7.9

86.6

4.9

51.3

50

422

80

656

Mean of line 1 =

8.66

Mean of line 2 =

5.13

Mean of angle 1 =

42.2

Mean of angle 2 =

65.6

...read more.

Conclusion

           _______      _____

SD= √1649/20 = √82.45 = 9.08 to 2 d.p

Angle 2

           __________    _______

SD= √1820.55/20 =√91.0275 = 9.54 to 2d.p

        As the standard deviation is smaller, the results are closer to the actual number. The standard deviation that I have obtained shows that people did guess the length of lines much better than the size of the angle because the standard deviation for the length of lines was much smaller than it is for the size of the angle. This shows that my hypothesis was correct and that it is better to use standard deviation than it is to use the mean. It would also be better than using the mode and the median because unlike them, standard deviation takes in account all of the results that have been obtained. This also shows me that the comparison of 1 mm to 1 degree is not a very good comparison as it showed me the opposite to what standard deviation did. So in conclusion, both of my hypotheses were proved to be true by using the mean and standard deviation as they both take in account all the results so they are the most accurate. Standard deviation is more accurate than using the mean. Although they are both taking in account all of the results, standard deviation allows you to compare any two or more things with different units e.g. centimetres and degrees. Standard deviation does not have a unit so it is much more accurate to compare than it is to the mean. With the mean you have to compare the units and it is very hard when the units are not comparable e.g. centimetres and degrees. When using the mean you will be obliged to make a comparison for the two results like the 1mm to 1º comparison I made. Sometimes these comparisons can be very inaccurate like I have discovered which gives wrong results. That is why it is better to use standard deviation for comparison that it is to use the mode, median, mean, range or inter quartile range.

...read more.

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