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&quot;The lengths of lines are easier to guess than angles. Also, that year 11's will be more accurate at estimating.&quot;

Extracts from this document...

Introduction

Hisham Band                Maths Coursework – “Guesstimate”

In this investigation, 3 year groups – years 9, 10 and 11, were asked to estimate the lengths of some lines and angles, and the results that the pupils produced are going to be analysed to try and prove or disprove the hypothesis of:

“The lengths of lines are easier to guess than angles. Also, that year 11’s will be more accurate at estimating.”

The reasons I think these things are because people are more used to seeing lines than they are angles, so this could mean that they are better at estimating the length of lines. The reason I think they year 11’s will be more accurate is because they have done maths longer than the year 9’s, so they have had more experience.

I will be using an example of one line, and one angle, and the results of Year 9 and Year 11 estimates. This is secondary data which has been previously recorded, during a survey to find out the estimates that the pupils gave. This data is continuous as it is As there are 117 year 9’s and 145 year 11’s I will have to reduce the size of my sample as these numbers are too large to handle, so I will be using a stratified method to reduce the size of the samples as this method keeps the results for the year groups in proportion to each other.

I am going to be sampling 60 people in total, out of the year 9’s and year 11’s, as this is a manageable amount, and it can represent the data from the two year groups accurately as a smaller number might not show the difference in results suitably.

To choose my samples I am first going to add together the two total numbers of each year group, which is:

145 + 117 = 262  (Year 11 / Year 9)

Middle

1.4

30.43

40

7

21.21

4

-0.6

-13.04

30

-3

-9.09

4

-0.6

-13.04

20

-13

-39.39

5

0.4

8.70

45

12

36.36

5

0.4

8.70

40

7

21.21

5

0.4

8.70

35

2

6.06

4

-0.6

-13.04

35

2

6.06

6

1.4

30.43

40

7

21.21

4.6

0

0.00

30

-3

-9.09

5

0.4

8.70

35

2

6.06

4.5

-0.1

-2.17

40

7

21.21

4.5

-0.1

-2.17

45

12

36.36

3.5

-1.1

-23.91

43

10

30.30

3.5

-1.1

-23.91

43

10

30.30

3

-1.6

-34.78

50

17

51.52

4.5

-0.1

-2.17

45

12

36.36

6

1.4

30.43

30

-3

-9.09

5

0.4

8.70

35

2

6.06

5

0.4

8.70

45

12

36.36

Total

6.6

143.48

Total

238

721.21

The mean percentage error is worked out by dividing the total percentage error by how many pieces of data there are. So, for the mean of the year 9 line errors, the calculation would be 143.48/27 which equals 5.31%. This is quite a low percentage of error which means that the year 9’s were quite good at estimating the line. For the angle it would be 721.21/27 which equals 26.71%. This percentage is so high because someone estimated 95º, which means that mean is made higher by this anomalous result. This shows that the year 9’s were better at estimating the length of the line.

The next table is for the  percentage error of year 11 estimates.

LINE                                                ANGLE

 Estimate (e) Error Percentage error (%) Estimate (e) Error Percentage error (%) 5 0.4 8.70 40 7 21.21 5 0.4 8.70 35 2 6.06 5 0.4 8.70 40 7 21.21 4 -0.6 -13.04 45 12 36.36 5 0.4 8.70 45 12 36.36 5 0.4 8.70 45 12 36.36 5 0.4 8.70 45 12 36.36 3 -1.6 -34.78 40 7 21.21 5 0.4 8.70 46 13 39.39 5 0.4 8.70 30 -3 -9.09 4 -0.6 -13.04 30 -3 -9.09 3.5 -1.1 -23.91 40 7 21.21 4.5 -0.1 -2.17 29 -4 -12.12 5 0.4 8.70 45 12 36.36 4 -0.6 -13.04 30 -3 -9.09 4.5 -0.1 -2.17 30 -3 -9.09 3 -1.6 -34.78 30 -3 -9.09 4 -0.6 -13.04 35 2 6.06 8 3.4 73.91 40 7 21.21 4 -0.6 -13.04 30 -3 -9.09 8 3.4 73.91 40 7 21.21 4 -0.6 -13.04 45 12 36.36 6 1.4 30.43 45 12 36.36 5 0.4 8.70 45 12 36.36 5 0.4 8.70 35 2 6.06 5 0.4 8.70 50 17 51.52 4 -0.6 -13.04 45 12 36.36 4 -0.6 -13.04 35 2 6.06 4 -0.6 -13.04 45 12 36.36 5 0.4 8.70 35 2 6.06 4.5 -0.1 -2.17 40 7 21.21 5 1.4 30.43 45 12 36.36 6 1.4 30.43 40 7 21.21 Total 6.2 134.78 Total 206 624.24

The mean for the above table of the line errors would be 134.78/33 which equals 4.08%. This is a relative low percentage which means from the mean there doesn’t seem to be much error for the line. For the angle the mean is 624.24/33 which equals 18.92%. This means that from the mean you can see that from the year 11 data, they were better at estimating the length of the line.

From these calculations, I have found that the year 11’s were better at estimating by finding the mean percentage error, as both of the mean percentage errors for the line and angle, were lower than the year 9 errors.

From the percentage error tables I can plot a scatter graph to see how much error there was for the angle compared to the line for both year groups. These graphs were drawn from the actual error made. For example, if someone’s error for the line was -2, because they guessed 2 below the actual angle size, and their error for the angle was 3, because they guessed 3 above the actual length of the line, then the co-ordinate for their error would be (-2,3).

This scatter graph shows the error of the year elevens. If someone had estimated exactly right and therefore had no error then there mark would be at the point (0,0) on the scatter graph. The point circled in pink is an anomalous result. This is a result which is out of the pattern of the rest of the results.

This scatter graph shows the error of the year 9 estimates. The pink circled result is again an anomalous result.

Stem and leaf diagrams put the data in numerical order in an easy to read table. This is a stem and leaf diagram to show Angle estimates:

 Year 9 ANGLE 6 Year 11 0 2 9 5,5,5,5,5,5,5,0,0,0 3 0,0,0,0,0,0,5,5,5,5,5 5,5,5,5,3,3,0,0,0,0,0 4 0,0,0,0,0,0,0,0,5,5,5,5,5,5,5,5,5,5,5,6 3,0,0 5 0 0 6 7 8 5 9

KEY  0   2   9 means that the year 9 estimate was 20 and the year 11 estimate was 29.

This is a stem and leaf diagram to show the Line estimates:

 YEAR 9 LINE 2 YEAR 11 5,5,0 3 0,0,5 6,5,5,5,5,0,0,0,0 4 0,0,0,0,0,0,0,0,0,5,5,5 0,0,0,0,0,0,0,0 5 0,0,0,0,0,0,0,0,0,0,0,0,0,0 2,0,0,0,0,0,0 6 0,0 8 0,0

Conclusion

Additionally, from the scatter graphs I was able to see that there were some anomalous results. For example there was a year 9 who estimated the angle at being 95º, whereas they estimated the line at being 5cm which showed that they did not estimate too extremely for the length of the line. I do not feel these scatter graphs assisted me in proving my hypothesis.

Overall, I feel I have been able to prove this hypothesis as correct through the calculations which referred to it.

I think all of my calculations and diagrams were correct as they all led to the same conclusion, and through checking my answers I found that they were correct. I think there were not really any major anomalous results, apart from the ones previously mentioned and shown in my calculations.

One problem I did have was finding the standard deviation, but I realised this was because I was trying to find it from grouped data, and therefore needed a different formula, and after getting the formula for grouped data I found it a lot easier to do.

I managed to prove my hypotheses were correct in most instances and I managed to show many different types of calculations in proving my hypotheses.

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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