- Level: AS and A Level
- Subject: Maths
- Word count: 3335
"Males in the 11-18years age range will guess the angles and lengths better than females in the 30+years age range."
Extracts from this document...
Introduction
INTRODUCTION:
We have been given a task to work with the statistical side of maths.
We were told that a random sample of one hundred and fifty people were asked to estimate the degree of an angle which looked like this:
Also each person was asked to estimate the length of a line which looked something like this:
From this we were asked to think about what sort of conclusions we could give for the analysed results. I thought about this and came up with the following hypothesis:
“Males in the 11-18years age range will guess the angles and lengths better than females in the 30+years age range.”
I have chosen this hypothesis because it allows me to look at both age and gender of the surveyed people.
(The actual measurement of the angle was 36.5degrees and the actual measurement of the line was 4.35cm)
The data I have received from the survey is on the following page…
PLAN:
To investigate this hypothesis I may need to use a number of methods to calculate the data I will receive. I may need to use many statistical tools which I already know; such as Averages, Standard Deviation, Frequency Polygons, Histograms, Scatter Diagrams, Cumulative frequency and maybe other methods which I will have to research.
As the sample I am being given is random, it has a large range of varied aged males and varied aged females. I need to divide the groups up into the groups I have to work with according to my hypothesis.
Middle
12
50
8
7
36
m
12
45
3
8
38
m
12
35
4
9
41
m
12
45
4
10
43
m
13
30
4.5
11
50
m
13
30
5
12
51
m
13
45
4.3
13
53
m
13
40
3
14
56
m
13
40
4.5
15
118
m
15
45
4
16
120
m
15
45
5
17
127
m
15
40
4
18
130
m
15
45
4
19
132
m
15
40
4
20
148
m
16
40
5
Females aged 30+years:
No. | Gender | Age | Angle est (deg) | Length est (cm) | |
1 | 70 | f | 61 | 30 | 4 |
2 | 74 | f | 56 | 40 | 3 |
3 | 77 | f | 54 | 37 | 5 |
4 | 79 | f | 51 | 30 | 3.5 |
5 | 80 | f | 50 | 40 | 3 |
6 | 84 | f | 49 | 45 | 3.5 |
7 | 85 | f | 48 | 43 | 3.3 |
8 | 87 | f | 47 | 45 | 5 |
9 | 89 | f | 45 | 40 | 2.5 |
10 | 90 | f | 45 | 37 | 3.2 |
11 | 91 | f | 44 | 45 | 3 |
12 | 92 | f | 43 | 45 | 4.5 |
13 | 93 | f | 43 | 40 | 3.2 |
14 | 96 | f | 42 | 45 | 4.5 |
15 | 97 | f | 42 | 45 | 5.1 |
16 | 99 | f | 41 | 45 | 5 |
17 | 101 | f | 38 | 39 | 5.3 |
18 | 106 | f | 35 | 45 | 5 |
19 | 108 | f | 31 | 45 | 7 |
20 | 110 | f | 30 | 35 | 10 |
These samples are now easier to work with and compare. I can now use these samples to make Frequency polygons of angle and length estimates of each of the groups.
Angle estimates of the male 11-18years sample:
Length estimates of the male 11-18years sample:
Angle estimates of the female 30+years sample:
Length estimates of the female 30+years sample:
The magenta line on each of the frequency polygons shows the actual length/angle. The nearer the navy blue line (showing the estimation) is to the magenta line, the more accurate the estimate was. You can clearly see that some people’s estimates were way off while others were quite near. You can see that nobody from the samples estimated exactly (though the chances of that are next to 0). The correlation of the blue line in respect to the magenta line shows how accurate the group was as a whole – i.e. the more the blue line is closer to the magenta line, the more accurate the group as a whole was. Or if the points of the blue line are all really far from the magenta line, we can say that that group was inaccurate.
In terms of the four polygons I have compiled here, it is hard to say if a group is accurate or not as even one estimate can completely sway the frequency out of proportion. The frequency polygon labelled ‘Angle estimates of the male 11-18years sample’ shows that most of the estimates were quite far off being right although two of the estimates are very close, which gives me something to consider. The frequency polygon labelled ‘Length estimates of the male 11-18years sample’ the correlation of the blue line is quite good as the points are relatively close to the magenta line but either side of it, showing that the estimates are close overall. Only one estimate from this polygon is way off. The female frequency polygons are relatively the same as he male ones in terms of correlation (though one estimate of the length was a long way off). This shows me generally that males of 11-18 years estimate at approximately the same accuracy of females of 30+years, therefore disproving my hypothesis. From I can also say that, on average, people find it easier to judge the length of a line than to estimate an angle.
I still need to investigate further. Another way in which I can find out how accurate the groups in general are is to work out how accurate the male and female groups are on average. Therefore we must work out an average result for the length and the angle in each group. I will now work out the mean angle and length for each group. This is simply the sum of all the angles or lengths divided by twenty.
Males aged 11-18years:
Average estimated Length: Σl/20 = 4.44cm
Average estimated angle: Σa/20 = 40.9degrees
Females aged 30+years:
Average estimated Length: Σl/20 = 4.43cm
Average estimated angle: Σa/20 = 40.8degrees
(where: Σ = sum of, l = length, a = angle)
N.B. – Actual Length of line = 4.35cm
Actual Size of Angle = 36.5º
This disproves my hypothesis further by further backing up what I said before: ‘This shows me generally that males of 11-18 years estimate at approximately the same accuracy of females of 30+years’. You can see that the average estimates are almost identical. Also, the average estimates of the length of the line are pretty close to the actual length of the line but the average estimates of the angle were quite a bit off, but both male and female groups got approximately the same average estimate so I can’t really say that one group is better at estimating than the other.
I decided that this was not enough evidence to disprove my hypothesis properly so I researched into other methods of data representation and means of calculating the data. I came across ‘Spearman's Rank Correlation Coefficient’. The following information within the red boxes on the next three pages is what I researched (including examples):
Using the Spearman's Rank Correlation Coefficient I can give each of my samples a value to see how accurate they were more precisely in comparison. To do this however, I need to have something to rank. I will rank how close each person was to the actual measurement. To do this I will have to do Length estimate - 4.35 (or 4.35 – Estimate Length, depending on if the estimate is more or less than the actual measurement) and Angle estimate - 36.5 (or 36.5 – Estimate Angle, depending on if the estimate is more or less than the actual measurement.)
Males in the 11-18years sample (showing size of errors):
No. | Gender | Age | Angle est (deg) | Length est (cm) | Angle error (deg) | Length error (cm) | Error (A+L) |
1 | m | 14 | 45 | 4 | 8.5 | 0.35 | 8.85 |
2 | m | 14 | 30 | 4.5 | 6.5 | 0.15 | 6.65 |
18 | m | 15 | 47 | 5 | 10.5 | 0.65 | 11.15 |
19 | m | 14 | 45 | 5 | 8.5 | 0.65 | 9.15 |
22 | m | 12 | 36 | 4 | 0.5 | 0.35 | 0.85 |
30 | m | 12 | 50 | 8 | 13.5 | 3.65 | 17.15 |
36 | m | 12 | 45 | 3 | 8.5 | 1.35 | 9.85 |
38 | m | 12 | 35 | 4 | 1.5 | 0.35 | 1.85 |
41 | m | 12 | 45 | 4 | 8.5 | 0.35 | 8.85 |
43 | m | 13 | 30 | 4.5 | 6.5 | 0.15 | 6.65 |
50 | m | 13 | 30 | 5 | 6.5 | 0.65 | 7.15 |
51 | m | 13 | 45 | 4.3 | 8.5 | 0.05 | 8.55 |
53 | m | 13 | 40 | 3 | 3.5 | 1.35 | 4.85 |
56 | m | 13 | 40 | 4.5 | 3.5 | 0.15 | 3.65 |
118 | m | 15 | 45 | 4 | 8.5 | 0.35 | 8.85 |
120 | m | 15 | 45 | 5 | 8.5 | 0.65 | 9.15 |
127 | m | 15 | 40 | 4 | 3.5 | 0.35 | 3.85 |
130 | m | 15 | 45 | 4 | 8.5 | 0.35 | 8.85 |
132 | m | 15 | 40 | 4 | 3.5 | 0.35 | 3.85 |
148 | m | 16 | 40 | 5 | 3.5 | 0.65 | 4.15 |
Conclusion
Spearman's Rank Correlation Coefficient work I just did, I conclude that gender has near to no relationship with the amount of error made in guessing.
EVALUATION:
I think that the work I have done to investigate my hypothesis was relevant enough to disprove it with enough evidence. The method of sampling I chose was very useful and was the best form of sampling to use in my case and broke the data down in suitable, manageable parts. Of course, I could have made my stratas larger to get more accurate data, but 20 from each group was enough to work with and get a sensible result to draw a conclusion from. From the two groups, I selected 20 people from each at random, I think that if I had selected different people my results may have turned out differently (though not that much because there was not many to randomly select from in the first place, so many of the ones I have used would be used anyway). Looking over my work I do not think I have made too much of an error though in the course of calculations I did make some errors which I quickly corrected. I could have gone further into the investigation by using Spearman's Rank Correlation Coefficient more to find out the relationships between other aspects which are relevant to my hypothesis, though I felt that I had enough evidence to prove my hypothesis. Also I could have used more of the statistical tools I stated in the plan to get an even broader approach to investigating my hypothesis.
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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