I am investigating how well people estimate the length of a line and the size of an angle.
I am investigating how well people estimate the length of a line and the size of an angle. I am going to compare the following:
* Year 7 compared to year 10 (Boys and girls) in estimating the size of an acute angle.
* Girls compared to boys (Years 7 and 10) in estimating the length of a short line.
I am going to compare these two because it is a very wide range of data. I am going to sample 40 people for each investigation,
For example:
* 40 people from year 7 and,
* 40 people from year 10.
A questionnaire has been circulated to a variety of people in set 1-5 and year 7-sixth form. The questionnaire includes questions, such as:
* Estimate the length of this line
* Estimate the size of this angle
* Estimate the length of this squiggle
For Investigation 1
'Year 7 compared to year 10 in estimating the size of an acute angle'
My hypothesis is that a larger amount of year 10 will be better at estimating the size of an angle than year 7. I think that more people from year 10 will be better at estimating the size of an angle because they have been in education longer and are more advanced at maths, while year 7 will be less advanced as they haven't been in education as long as year 10.
For Investigation 2
'Girls compared to boys I estimating the length of a short line'
My hypothesis is that girls will be better at estimating the length of a line and the size of an angle than boys. I think that girls will be better at estimating the length of a line and the size of an angle than boys because girls take time to study things more closely than boys as boys tend to rush things.
I will put all the data I collect in graphs. I will put my data in the following graphs:
* Box plot
* Cumulative frequency graph
I have chosen these graphs because they are clear, simple to look at and will show my results best. A box plot provides an excellent visual summary of important aspects of a distribution among my data. The box stretches from the lower quartile to the upper quartile and therefore contains the middle half of the data in the distribution. The median is shown as a line across the box. Therefore 1/4 of the distribution is between this line and the top of the box and 1/4 of the distribution is between this line and the bottom of the box. This makes it easy for me to comment on my data and compare two sets of data if they are both in box plots.
From the cumulative frequency graph it is possible to work out three important statistics:
* The lower and upper quartiles
* The median
* The interquartile range
From these 3 important statistics I will find it easy to compare data both on other cumulative frequency graphs and on box plots making it easier for me to come to a conclusion about my data and find out whether or not my hypothesis is correct.
I will use stratified random sampling to sample my data because it is an alternative to a simple random sample that provides more precision. In a simple random sample, I would select subjects randomly from a single large pool of data. In a stratified random sample, I will divide this large pool of subjects into several groups called strata (in this case strata will be gender and year group) and then randomly select subjects from within each group. The number of subjects selected from each group is fixed by design.
A stratified sample makes sense when your data is varied, but it can easily be split into strata that are more consistent. I am using a stratified sample because there is a lot variability between strata and little variability within strata.
The numbers I select from each strata will be proportional to the size of the strata.
CALCULATIONS
*The numbers I select from each strata will be proportional to the size of the strata, I am sampling 20% of each strata. I will then make this fair by finding the mean average of each strata with will in turn help me find out whether or not my hypothesis is right*
Hypothesis 1:
Year 10 will be better at estimating the size of an acute angle than Year 7:
Method: I will select my data of people's estimates for usage in hypothesis 1 and 2 using stratified random sampling. I intend to do this by sampling 20% of each strata, in these cases gender and year group. Once I have worked out what proportion of each strata I need to sample I will use the 'random' button on my calculator to randomly sample my data, to avoid biased data. I will then work out how far out their estimate was off the actual measurement and from that I will work out the percentage error so I can work out whether my hypothesizes are correct. I will work out whether my hypothesizes are correct by working out what the average percentage error is for each group. I will list the estimate, error and percentage error in order of ascending size to make it easier for myself when adding up percentage errors and dividing them by the amount of data in that particular strata, to get the mean average.
Then I will for hypothesis 1(Year 10 will be better at estimating the size of an acute angle than Year 7):
* Add together the mean average of males and females in year 7 and divide it by 2 to get the mean average of males and females In year 7 combined.
* Add together the mean average of males and females in year 10 and divide it by 2 to get the mean average of males and females In year 10 combined.
* Compare the 2 averages. If the average for year 10 is ...
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Then I will for hypothesis 1(Year 10 will be better at estimating the size of an acute angle than Year 7):
* Add together the mean average of males and females in year 7 and divide it by 2 to get the mean average of males and females In year 7 combined.
* Add together the mean average of males and females in year 10 and divide it by 2 to get the mean average of males and females In year 10 combined.
* Compare the 2 averages. If the average for year 10 is a lower percent than that of year 7, my hypothesis is correct., as this means the average percentage error is lower for year 10 than year 7, therefore meaning, on average, year 10 were more accurate and made less mistakes. If the average for year 7 is a lower percent than that of year 10, my hypothesis is incorrect., as this means the average percentage error is lower for year 7 than year 10, therefore meaning, on average, year 7 were more accurate and made less mistakes
Then I will for hypothesis 2 (Girls will be better at guessing the size of a short line than boys):
* Add together the mean average of females in year 7 and year 10 and divide it by 2 to get the mean average of females in year 7 and 10 combined.
* Add together the mean average of males in year 7 and year 10 and divide it by 2 to get the mean average of males in year 7 and 10 combined.
* Compare the 2 averages. If the average for females is a lower percent than that of males, my hypothesis is correct., as this means the average percentage error is lower for year females than males, therefore meaning, on average, females were more accurate and made less mistakes. If the average for males is a lower percent than that of females, my hypothesis is incorrect., as this means the average percentage error is lower for year males than females, therefore meaning, on average, males were more accurate and made less mistakes
Year 7 Females
Stratified random sampling:
00 year 7 females in total.
20% of 100= 20. I will therefore take 20 samples.
Estimate
Error
% error
32
3
30
3
9
25
8
24
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
20
3
39
20
3
39
49
6
48
5
8
55
60
27
82
60
27
82
65
32
97
Average Angle percentage error for Girls in year 7:
3
9
24
36
36
36
36
36
36
36
36
36
36
36
36
39
39
48
55
82
82
+97
910
910- 20= 45.5%
Year 7 Males
Stratified random sampling:
10 year 7 males in total.
20% of 110= 22. I will therefore take 22 samples.
Estimate
Error
% error
32
3
30
3
9
25
8
24
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
20
3
39
20
3
39
49
6
48
5
8
55
60
27
82
60
27
82
65
32
97
Average Angle percentage error for Boys in year 7:
6
6
6
9
9
9
21
21
21
21
36
36
36
36
36
36
39
39
100
100
100
+100
844
844-22= 38.36%= 38.4%
Year 10 Females
Stratified random sampling:
80 year 10 females in total.
20% of 80= 16. I will therefore take 16 samples.
Estimate
Error
% error
35
2
6
30
3
9
30
3
9
30
3
9
30
3
9
30
3
9
40
7
21
40
7
21
40
7
21
40
7
21
40
7
21
43
0
30
45
2
36
45
2
36
45
2
36
45
2
36
Average Angle percentage error for Girls in year 10:
6
9
9
9
9
9
21
21
21
21
21
30
36
36
36
+36
330
330- 16= 20.625%= 20.6%
Year 10 Males
Stratified random sampling:
00 year 10 males in total.
20% of 100= 20. I will therefore take 20 samples.
Estimates
Error
% error
35
2
6
30
3
9
30
3
9
38
5
5
27
6
8
40
7
21
40
7
21
40
7
21
40
7
21
40
7
21
40
7
21
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
20
3
39
50
7
52
50
7
52
60
27
82
Average Angle percentage error for Boys in year 10:
6
9
9
15
18
21
21
21
21
21
21
21
21
36
36
36
36
36
39
52
52
+82
630
630- 20= 31.5%
Year 7 girls: 45.5%
Year 7 boys: 20.625%
45.5
+20.625
66.125
66.125- 2= 33.0625= 33.1%(1dp) Year 7 percentage error, as used for hypothesis 1 (Year 10 will be better at estimating the size of an acute angle than Year 7) If this is lower than the percentage error of year 10 (as below) my hypothesis will be proved incorrect, whereas if it is higher than that of year 10 (as below) my hypothesis will be proved correct.
Year 10 girls: 38.36
Year 10 boys: 31.5
38.36
+ 31.5
69.86
69.86-2= 34.93= 34.9% Year 10 percentage error, as used for hypothesis 1 (Year 10 will be better at estimating the size of an acute angle than Year 7) If this is lower than the percentage error of year 7 (as above) my hypothesis will be proved correct, whereas if it is higher than that of year 7 (as above) my hypothesis will be proved incorrect.
MY HYPOTHESIS WAS PROVED INCORRECT AS ON AVERAGE THE PERCENTAGE ERROR FOR THE ESTIMATION OF AN ACUTE ANGLE WAS LOWER FOR YEAR 7 THAN FOR YEAR 10, MEANING YEAR 7 WERE MORE ACCURATE AND CLOSER TO THE CORRECT NUMBER (ON AVERAGE) THAN YEAR 10.
Hypothesis 2:
Girls will be better at guessing the size of a short line than boys.
Year 7 Females
Stratified random sampling:
00 year 7 females in total.
20% of 100= 20. I will therefore take 20 samples.
Estimates
Error
% error
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.45
0.05
3
.6
0.1
7
.6
0.1
7
.6
0.1
7
.7
0.2
3
.2
0.3
20
.2
0.3
20
.9
0.4
26
.9
0.4
26
2
0.5
33
0.5
33
0.2
.3
86
Average Short line percentage error for Girls in year 7:
3
7
7
7
13
20
20
26
26
33
86
464
464- 20= 23.2% average
Year 7 Males
Stratified random sampling:
10 year 7 males in total.
20% of 110= 22. I will therefore take 22 samples.
Estimate
Error
% error
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.6
0.1
7
.25
0.25
7
.2
0.3
20
.2
0.3
20
.9
0.4
26
0.5
33
2
0.5
33
2
0.5
33
2.25
0.75
50
2.5
67
2.5
67
2.5
67
0.04
.46
97
3
.5
00
3.2
.7
13
3.4
.9
27
3.4
.9
27
3.4
.9
27
Average Short line percentage error for Boys in year 7:
7
17
20
20
26
33
33
33
50
67
67
67
97
100
113
127
127
127
131
131- 22= 51.40909= 51.4%
Year 10 Females
Stratified random sampling:
80 year 10 females in total.
20% of 80= 16. I will therefore take 16 samples.
Estimates
Error
% Error
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.6
0.1
7
.6
0.1
7
.3
0.2
3
.3
0.2
3
.7
0.2
3
.7
0.2
3
.8
0.3
20
.8
0.3
20
.9
0.4
26
0.5
33
2
0.5
33
Average Short line percentage error for Girls in year 10:
7
7
13
13
13
13
20
20
26
33
33
88
88- 16= 11.75%
Year 10 Males
Stratified random sampling:
00 year 10 males in total.
20% of 100= 20. I will therefore take 20 samples.
Estimate
Error
% error
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.6
0.1
7
.6
0.1
7
.7
0.2
3
.7
0.2
3
.2
0.3
20
.2
0.3
20
2
0.5
33
2
0.5
33
2
0.5
33
2.3
0.8
53
2.3
0.8
53
2.5
67
2.5
67
2.3
67
2.3
67
Average Short line percentage error for Boys in year 10:
7
7
13
13
20
33
33
33
53
53
67
67
67
+ 67
533
Girls:
Year 7 girls:23.2
Year 10 girls: 11.75
23.2
+11.75
34.95
34.95- 2= 17.475= 17.5%(1dp) Female percentage error, as used for hypothesis 2 (Girls will be better at guessing the size of a short line than boys) If this is lower than the percentage error of Males (as below) my hypothesis will be proved correct, whereas if it is higher than that of Males (as below) my hypothesis will be proved incorrect.
Boys:
Year 7 boys: 51.4
Year 10 boys: 26.65
51.4
+26.65
78.05
78.05-2=39.025 =39%(1dp) Male percentage error, as used for hypothesis 2 (Girls will be better at guessing the size of a short line than boys) If this is lower than the percentage error of Females (as above) my hypothesis will be proved incorrect, whereas if it is higher than that of females (as above) my hypothesis will be proved correct.
MY HYPOTHESIS WAS PROVED CORRECT AS ON AVERAGE THE PERCENTAGE ERROR FOR THE ESTIMATION OF A SHORT LINE WAS LOWER FOR FEMALES THAN FOR MALES, MEANING FEMALES WERE MORE ACCURATE AND CLOSER TO THE CORRECT NUMBER (ON AVERAGE) THAN MALES.
Method: I will display my data for both hypothesize in different graphs before drawing a final conclusion. I will display data from hypothesis 1 in a cumulative frequency table, then graph as I will find it easier to compare data both on other cumulative frequency graphs and on box plots, than I would do on perhaps a frequency polygon making it easier for me to come to a conclusion about my data and find out whether or not my hypothesis is correct. I will display data from hypothesis 2 on a box plot as it can be easily compared with other box plots and even cumulative frequency graphs so again, I can clearly see the data and it will be easier for me to reach a final conclusion.
Hypothesis 1 (Year 10 will be better at estimating the size of an acute angle than Year 7):
* I will sort (in ascending order) Year 7 Males and Females combined, to use when drawing out the cumulative frequency table.
* I will then (in ascending order) Year 10 Males and Females combined, to use when drawing out the cumulative frequency table.
* I will make cumulative frequency tables for both year 10 and year 7 individually, this will allow me to make a cumulative frequency graph, which is what I am using to help me conclude my results and check my hypothesis
* I will then draw cumulative frequency graphs for both hypothesizes. Giving me a total of 4 cumulative frequency graphs. On the 2 cumulative frequency graphs for the first hypothesis (Year 10 will be better at estimating the size of an acute angle than Year 7) I will write down the median, upper and lower quartiles I will do this by using the formula 1/2 (n+1) to work out the median (n being the cumulative frequency up the side of the graph). 1/4 (n+1) to work out the lower quartile and 3/4 (n+1) to work out the upper quartile. I will find these numbers along the graph, and down, so that I get a reading for median, and the upper and lower quartile boundaries.
* I will do as above for hypothesis 2. But I will then draw a box and whisker plot below the culumative frequency graph. I will do this by using the scale along the X axis and following down the median, and the upper and lower quartile boundaries. From this values I will join up in a box shape to show the interquartile range. I will then draw 'whiskers' to the smallest number in the group from one side of the box and from the other side, 'whiskers' to the highest number in the group.
Year 7, males and females, for hypothesis 1 (Year 10 will be better at estimating the size of an acute angle than Year 7) table of data in ascending order.
Estimate
Error
% error
32
3
32
3
30
3
9
30
3
9
25
8
24
25
8
24
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
20
3
39
20
3
39
20
3
39
20
3
39
49
6
48
49
6
48
5
8
55
5
8
55
60
27
82
60
27
82
60
27
82
60
27
82
65
32
97
65
32
97
Year 10, males and females, for hypothesis 1 (Year 10 will be better at estimating the size of an acute angle than Year 7) table of data in ascending order.
Estimate
Error
% error
35
2
6
35
2
6
30
3
9
30
3
9
30
3
9
30
3
9
30
3
9
30
3
9
30
3
9
38
5
5
27
6
8
40
7
21
40
7
21
40
7
21
40
7
21
40
7
21
40
7
21
40
7
21
40
7
21
40
7
21
40
7
21
40
7
21
43
0
30
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
45
2
36
20
3
39
50
7
52
50
7
52
60
27
82
Cumulative frequency table for Year 7 males and females for hypothesis 1 (Year 10 will be better at estimating the size of an acute angle than Year 7)
Group
Frequency
Cumulative Frequency
0-
4
4
0-
0
4
20-
2
6
30-
24
30
40-
2
32
50-
2
34
60-
0
34
70-
0
34
80-
4
38
90-
2
40
00-
0
40
Cumulative frequency:
40
/2(40+1)
/2 41
41-2= 20.5 Median
/4(40+1)
/4 41
41-4= 10.25 Lower quartile boundary
3/4(40+1)
3/4 41
(41-4) x3= 30.75 Upper Quartile Boundary
Cumulative frequency table for Year 10 males and females for hypothesis 1 (Year 10 will be better at estimating the size of an acute angle than Year 7)
Group
Frequency
Cumulative Frequency
0-
9
9
0-
2
1
20-
1
22
30-
1
33
40-
0
33
50-
2
35
60-
0
35
70-
0
35
80-
36
90-
0
36
00-
0
36
Cumulative frequency:
36
/2(36+1)
/2 37
37-2= 18.5 Median
/4(36+1)
/4 37
37-4= 9.25 Lower quartile boundary
3/4(36+1)
3/4 37
(37-4) x3= 27.45 Upper Quartile Boundary
Table leading onto box plot for hypothesis 2 (Girls will be better at guessing the size of a short line than boys)
Year 7 and 10 Females
Estimates
Error
% error
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.45
0.05
3
.6
0.1
7
.6
0.1
7
.6
0.1
7
.6
0.1
7
.6
0.1
7
.7
0.2
3
.3
0.2
3
.3
0.2
3
.7
0.2
3
.7
0.2
3
.2
0.3
20
.2
0.3
20
.8
0.3
20
.8
0.3
20
.9
0.4
26
.9
0.4
26
.9
0.4
26
2
0.5
33
0.5
33
0.5
33
2
0.5
33
0.2
.3
86
Table leading onto box plot for hypothesis 2 (Girls will be better at guessing the size of a short line than boys)
Year 7 and 10 Males
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.5
0
0
.6
0.1
7
.6
0.1
7
.6
0.1
7
.7
0.2
3
.7
0.2
3
.25
0.25
7
.2
0.3
20
.2
0.3
20
.2
0.3
20
.2
0.3
20
.9
0.4
26
0.5
33
2
0.5
33
2
0.5
33
2
0.5
33
2
0.5
33
2
0.5
33
2.25
0.75
50
2.3
0.8
53
2.3
0.8
53
2.5
67
2.5
67
2.5
67
2.5
67
2.5
67
2.3
67
2.3
67
0.04
.46
97
3
.5
00
3.2
.7
13
3.4
.9
27
3.4
.9
27
3.4
.9
27
Cumulative frequency table leading onto box plot for hypothesis 2 (Girls will be better at guessing the size of a short line than boys)
Year 7 and 10 Females
Groups
Frequency
Cumulative frequency
0-
9
9
0-
5
24
20-
7
31
30-
4
35
40-
0
35
50-
0
35
60-
0
35
70-
0
35
80-
36
90-
0
36
00-
0
36
Cumulative frequency:
36
/2(36+1)
/2 37
37-2= 18.5 Median
/4(36+1)
/4 37
37-4= 9.25 Lower quartile boundary
3/4(36+1)
3/4 37
(37-4) x3= 27.45 Upper Quartile Boundary
Cumulative frequency table leading onto box plot for hypothesis 2 (Girls will be better at guessing the size of a short line than boys)
Year 7 and 10 males
Groups
Frequency
Cumulative frequency
0-
2
2
0-
3
5
20-
5
20
30-
6
26
40-
0
26
50-
3
29
60-
7
36
70-
0
36
80-
0
36
90-
37
00-
38
10-
39
20-
3
42
Cumulative frequency:
42
/2(42+1)
/2 43
43-2= 21.5 Median
/4(42+1)
/4 43
43-4= 10.75 Lower quartile boundary
3/4(42+1)
3/4 42
(43-4) x3= 32.25 Upper Quartile Boundary
COUNCLUSION
Hypothesis 1:
MY HYPOTHESIS WAS PROVED INCORRECT AS ON AVERAGE THE PERCENTAGE ERROR FOR THE ESTIMATION OF AN ACUTE ANGLE WAS LOWER FOR YEAR 7 THAN FOR YEAR 10, MEANING YEAR 7 WERE MORE ACCURATE AND CLOSER TO THE CORRECT NUMBER (ON AVERAGE) THAN YEAR 10.
Hypothesis 2:
MY HYPOTHESIS WAS PROVED CORRECT AS ON AVERAGE THE PERCENTAGE ERROR FOR THE ESTIMATION OF A SHORT LINE WAS LOWER FOR FEMALES THAN FOR MALES, MEANING FEMALES WERE MORE ACCURATE AND CLOSER TO THE CORRECT NUMBER (ON AVERAGE) THAN MALES.
I have evidence in the form of graphs and average calculations to prove my conclusion. My hypothesis works on a base of average. I ruled out all anomalous results before stratifying my samples to avoid biased results which may lead me to conclude my hypothesizes wrong.
I think hypothesis 1 was incorrect because among Year 10 we had a few results which were a large percentage error (127%) this bought up my average and made my hypothesis worng.
I could improve my project and got better results my sampling more data so I got a more accurate average and by taking out all even slightly anomalous results to stop them from bringing the average percentage error up. To develop my task further, I could have gone on to look at another part of the questionnaire i.e. the estimation of a short squiggle.
