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# Estimation and testing of Hypotheses.

Extracts from this document...

Introduction

Lara Lavelle-Langham

Maths coursework

Estimation

## Contents

Aim,

Problem,

Hypotheses,

### Hypothesis one: -

Sample population and method,

Raw data,

Organisation of raw data

• Back to back stem and leaf diagram
• Cumulative frequency tables
• Cumulative frequency graphs (graph paper)
• Box-plot diagram (graph paper)
• Standard deviation

#### Averages

Male estimates/ female estimates

• Median
• Lower quartile
• Upper quartile
• Inter quartile range
• Mean
• Mode
• Range

Conclusion

### Hypothesis two: -

Raw data,

Organisation of raw data,

• Back to back stem and leaf diagram
• Histogram/frequency polygon tables
• Histogram graphs/frequency polygons (graph paper)

Averages

Under 18 estimates/ over 18 estimates

• Median
• Mean
• Mode
• Range

Conclusion

Evaluation

Plan:

Aim: to investigate and prove (or not be supported by the data obtained) two hypotheses.

Problem: Sarah asked students to estimate the length of a line and the size of an angle.

Using two hypotheses, design and carry out an experiment to test these.

Hypotheses: 1) Girls are better at estimating lengths than boys.

2) Boys over the age of 18 are better at estimating the size of an angle

than boys under 18.

First hypothesis

Sample population and method

When conducting the experiment I will attempt to ask every third person to estimate the sizes of the line and the angle.

Middle

5

0.41

0.1681

4.5

-0.09

0.0081

4.5

-0.09

0.0081

4.5

-0.09

0.0081

5

0.41

0.1681

5.5

0.91

0.8281

4.5

-0.09

0.0081

Sum of (estimate - mean)^2

8.687

Divide by 29

0.299551724

Root

0.547313187

Standard Deviation :

0.547313187

 Girls mean = 4.486666667 estimate estimate - mean (estimate - mean)^2 4.5 0.013333333 0.000177778 4.5 0.013333333 0.000177778 4.5 0.013333333 0.000177778 5 0.513333333 0.263511111 4.5 0.013333333 0.000177778 5 0.513333333 0.263511111 3.9 -0.586666667 0.344177778 4.5 0.013333333 0.000177778 4.8 0.313333333 0.098177778 5 0.513333333 0.263511111 4.5 0.013333333 0.000177778 4.5 0.013333333 0.000177778 4.9 0.413333333 0.170844444 4.3 -0.186666667 0.034844444 4.5 0.013333333 0.000177778 5 0.513333333 0.263511111 4.5 0.013333333 0.000177778 4 -0.486666667 0.236844444 4.5 0.013333333 0.000177778 4.5 0.013333333 0.000177778 4 -0.486666667 0.236844444 5.5 1.013333333 1.026844444 5 0.513333333 0.263511111 4.5 0.013333333 0.000177778 4.5 0.013333333 0.000177778 3.9 -0.586666667 0.344177778 4 -0.486666667 0.236844444 3.9 -0.586666667 0.344177778 3.9 -0.586666667 0.344177778 4 -0.486666667 0.236844444 Sum of (estimate - mean)^2 4.974666667 Divide by 29 0.17154023 Root 0.414174154 Standard Deviation : 0.414174154

Conclusion

When comparing the mean and the standard deviation of the estimates for the females and males I find that although the males mean is closer to the actual length of the line, the girls have a smaller standard deviation. This indicates that although further from the actual length of the line they are more consistent as a group.

The range for the females also supports this, it is a 0.5 cm difference between the two ranges.

Conceivably if the sample population had been given more specific guidelines for estimation of the line length then the results may have been more consistent and closer to the actual length. Examples of this may be to ask the sample population to estimate the length of the line to two decimal places or to specify that the line begins and ends before the ink from the other two lines closes the length.

Ultimately, I feel that I set out on this investigation without limiting enough of the degrees of freedom necessary to be able to point to the resultant evidence as either supporting or differing with the hypothesis.

## Hypothesis two

Males over the age of 18 are better at estimating the size of an angle than boys under 18.

## Data item

Age

Angle ( ° )

01

18

45

02

15

30

03

18

37

04

20

40

05

17

42

06

17

40

07

17

50

08

15

45

09

14

45

10

18

45

11

17

45

12

19

39

13

17

35

14

19

45

15

17

40

16

18

40

17

17

45

18

15

60

19

14

55

20

18

45

21

21

40

22

21

39

23

18

45

24

19

40

25

16

40

26

18

45

27

15

45

28

15

45

29

18

40

30

20

43

31

16

43

32

15

35

33

16

30

34

16

33

35

14

44

36

14

45

37

16

44

38

18

45

39

20

42

40

21

39

41

20

29

42

16

45

43

19

47

44

16

29

45

18

45

46

20

45

47

21

47

48

20

30

49

17

47

50

16

54

51

16

41

52

16

45

53

17

24

54

20

30

55

21

44

56

22

45

57

20

42

58

17

39

59

19

37

60

18

50

Conclusion

-1.5

2.25

18

40

-1.5

2.25

19

40

-1.5

2.25

20

40

-1.5

2.25

21

40

-1.5

2.25

20

42

0.5

0.25

20

42

0.5

0.25

20

43

1.5

2.25

21

44

2.5

6.25

18

45

3.5

12.25

18

45

3.5

12.25

18

45

3.5

12.25

18

45

3.5

12.25

18

45

3.5

12.25

18

45

3.5

12.25

18

45

3.5

12.25

19

45

3.5

12.25

20

45

3.5

12.25

22

45

3.5

12.25

19

47

5.5

30.25

21

47

5.5

30.25

18

50

8.5

72.25

Sum of (estimate - mean)^2

755.5

Divide by 29

26.05172414

Root

5.104088963

Standard Deviation :

5.104088963

Conclusion

Unfortunately the data collected is only marginally diverse. But from this there are data items that do show a fragment of variation between them.

The actual angle was 37.5°, although minute, the over 18’s were closer to the actual angle with a mean of 41.37° (2d.p). However both have different ranges with a difference of 15°.

The histograms showed the distribution of the frequencies. They do show a slight difference on the position of the modal class.

With the data and data presentation that I have I cannot in good confidence decide weather it invalidates or proves my hypothesis.

Evaluation

If I had the chance to repeat the investigation I would have liked to improve my hypotheses by making them more precise and accurate to allow maximum accuracy towards proving a hypothesis. This would have endorsed me to come to a solid conclusion.

As mentioned in the beginning of the investigation I only conducted the experiment inside the college, this did limit the age range of the sample population and may have contributed to the loss of a valid and inclusive conclusion.

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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