- Level: AS and A Level
- Subject: Maths
- Word count: 4461
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.
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 |
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.
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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