# We are conducting an investigation to discover how accurate people in our school are at estimating the lengths of lines and the sizes of angles.

Extracts from this document...

Introduction

Samantha Brown 11C

## Statistics Coursework

We are conducting an investigation to discover how accurate people in our school are at estimating the lengths of lines and the sizes of angles. My hypothesis will be based on sizes of lines and angles because if we tested people by using measurements like area and volume we may instead end up testing their ability at calculating these things instead. Some people find calculating these values harder than others.

### Hypothesis

- It is easier to estimate the size of acute angles than obtuse.

I think that it is easier to estimate the size of acute angles than obtuse angles because they are easier to visualise than larger angles. For example, the angles 90˚ and 45˚ are easier to visualise than angles like 165˚ and 130˚.

- Year 11 students can estimate the length of the straight lines more accurately than Year 7 students.

I have made this hypothesis because I think that Year 11 students have had more experience at using and estimating lengths due to the fact that they are 4 years older than year 7 students.

#### Method

I will be using a total of 60 students from which to collect the data needed for this investigation, 50% from Year 11 and 50% from Year 7. (However only 30 people from year 11 will be used for the investigation into my 1st hypothesis). This way I can compare the results I get from each year and either prove or disprove my hypotheses. I am using students from Years 7 and 11 because I believe this will make the difference between my results very different and will also show how we progress from Year 7 through to Year 11.

Two angles will be drawn onto a blank piece of A4 paper-one measuring 52˚ the other 156˚. These will be used to test candidates for my first hypothesis. A straight line measuring 13.

Middle

29(M)

63

11

150

6

19(M)

48

4

143

13

37(M)

49

3

169

13

5(M)

50

2

158

2

11(M)

55

3

153

3

1(M)

45

7

132

24

65(M)

47

5

147

9

57(M)

57

5

156

0

48(M)

45

7

166

10

52(M)

52

0

130

26

60(M)

50

2

174

18

21(M)

46

6

143

13

40(M)

55

3

175

19

17(M)

50

2

158

2

Mean

51.7666667

4.43333333

148.83333

14.233333

Mode

50

2

143

13

Hypothesis 1 Results Table Analysis

I think that overall the results I recorded for my first hypothesis support my idea that it is easier to estimate the size of an acute angle than an obtuse angle. The mean estimate of the acute angle (which was actually 52˚) came to about 52˚. The mean estimate of the obtuse angle (which was actually 156˚) came to about 149˚. The mean absolute error for the acute angle came to about 4˚, and the mean absolute error for the obtuse angle came to about 14˚. I think that the mean values for the absolute error of both angles back up my hypothesis the best, as they show a larger range (10˚) than errors shown when mean values of the estimates were worked out (about 3˚). I don’t think that the mean estimates give a very good idea of the accuracy of estimating as it is worked out using a large range of numbers (acute angle: 45˚-64˚, obtuse angle: 27˚-175˚), but the fact that the mean actual estimate for the acute angle is the same as the actual value gives strong evidence that my hypothesis is probably true. However, I believe the mean absolute errors to be more accurate as there is a smaller range of numbers used in the calculation, (acute angle: 0˚-12˚, obtuse angle: 0-129˚).

The mean values I have calculated for the obtuse angle will have been greatly affected by the anomalous result of 27˚, which I have highlighted in my results table. When the anomalous result is left out of the calculations the mean estimate becomes about 153˚, and the mean absolute error becomes about 10˚. This seems to make my hypothesis a little less true as the margin of error becomes smaller. However, I think that the fact that an anomalous result was recorded as an obtuse angle estimate backs up my hypothesis even more. It probably occurred because the candidate found it very difficult estimating the obtuse angle, as their estimate for the acute angle was only 3˚ out.

The modal values for each angle size also back up my hypothesis, as the errors for the obtuse angle are much greater than those of the acute angle. (The difference between the 2 absolute error modal values is 11˚).

## Hypothesis 2 Results | |||||

Year 7 Results | Year 11 Results | ||||

Student (F=Female, M=Male) | Estimate for line (cm) | Absolute Error (cm) | Student (F=Female, M=Male) | Estimate for line (cm) | Absolute Error (cm) |

36 (F) | 13.5 | 0.2 | 12(F) | 14 | 0.3 |

27 (F) | 10 | 3.7 | 49(F) | 13.5 | 0.2 |

25 (F) | 14 | 0.3 | 8(F) | 10 | 3.7 |

33 (F) | 13 | 0.7 | 21(F) | 13 | 0.7 |

4 (F) | 9 | 4.7 | 45(F) | 14 | 0.3 |

50(F) | 13.5 | 0.2 | 16(F) | 11 | 2.7 |

43 (F) | 9.5 | 4.2 | 23(F) | 14 | 0.3 |

47 (F) | 10 | 3.7 | 1(F) | 13 | 0.7 |

29 (F) | 8 | 5.7 | 46(F) | 13.5 | 0.2 |

18 (F) | 13 | 0.7 | 30(F) | 10.5 | 3.2 |

3 (F) | 8.5 | 5.2 | 50(F) | 11.5 | 2.2 |

16 (F) | 10 | 3.7 | 13(F) | 13 | 0.7 |

22 (F) | 15 | 1.3 | 9(F) | 13 | 0.7 |

1 (M) | 16 | 2.3 | 23(M) | 14 | 0.3 |

21 (M) | 13.5 | 0.2 | 18(M) | 14.2 | 0.5 |

10 (M) | 14 | 0.3 | 47(M) | 11.5 | 2.2 |

28 (M) | 8 | 5.7 | 29(M) | 13.5 | 0.2 |

16 (M) | 14 | 0.3 | 19(M) | 15 | 1.3 |

26 (M) | 11 | 2.7 | 37(M) | 13.5 | 0.2 |

58 (M) | 11.5 | 2.2 | 5(M) | 11 | 2.7 |

41 (M) | 12.5 | 1.2 | 11(M) | 15 | 1.3 |

42 (M) | 13.5 | 0.2 | 1(M) | 14 | 0.3 |

66 (M) | 9.5 | 4.2 | 65(M) | 13 | 0.7 |

4 (M) | 11.5 | 2.2 | 57(M) | 11 | 2.7 |

30 (M) | 8 | 5.7 | 48(M) | 10 | 3.7 |

9 (M) | 7.5 | 6.2 | 52(M) | 14 | 0.3 |

65 (M) | 16 | 2.3 | 60(M) | 14.5 | 0.8 |

39 (M) | 10.5 | 3.2 | 21(M) | 13.5 | 0.2 |

63 (M) | 11 | 2.7 | 40(M) | 10.5 | 3.2 |

69 (M) | 8.5 | 5.2 | 17(M) | 14 | 0.3 |

Mean | 11.45 | 2.7033333 | Mean | 12.873333 | 1.2266667 |

Mode | 13.5 | 0.2 | Mode | 14 | 0.3 |

Conclusion

Although there were some flaws in my data collection which may have caused bias, I believe that my investigation was a success and gave a fair representation of the sample groups used. The sample size of 60 students - 30 from Year 11, 30 from Year 7 - that I used was adequate enough to collect the results I needed to support my hypothesis in a fair and unbiased way. I think this was due to the careful planning and preparation that went into my method and sampling technique.

My results all seemed to back up my hypothesis and I managed to analyse my results thoroughly enough to explain how I believed them to support or refute my theories.

Scatter Graph Analysis

On this scatter graph instead of drawing a line of best fit I drew lines where the actual values of the obtuse and acute angles were. This has proved to be good visual aid when comparing the accuracy of the acute and obtuse angles.

The majority of crosses are below the 52˚ line. This shows that most people underestimated the size of the acute angle. The same can be seen in the obtuse angle estimates as most crosses are to the left of the 156˚ line. This shows that the size of the obtuse angle was also underestimated by most students. Although this shows inaccuracy in both sets of estimates, I believe that the estimates for the acute angle were more accurate than those for the obtuse angle as the majority of crosses appear to be closer to the 52˚ line.

Although I believe that this graph backs up my hypothesis, I do recognise the fact that it isn’t as clear to see as I had first hoped, and it also demonstrates that only one person estimated the correct number for each angle size. This limitation could be resolved if the scale on each axis of my scatter graph was the same.

This student written piece of work is one of many that can be found in our AS and A Level Probability & Statistics section.

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