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A class of students had a sudoku competition - Edexcel - A-Level Maths Statistics - Question 5 - 2011 - Paper 2
Question 5
A class of students had a sudoku competition. The time taken for each student to complete the sudoku was recorded to the nearest minute and the results are summarise... show full transcript
Worked Solution & Example Answer:A class of students had a sudoku competition - Edexcel - A-Level Maths Statistics - Question 5 - 2011 - Paper 2
Step 1
Step 2
Use linear interpolation to estimate the median time taken by the students.
Answer
To find the median, we need to determine the position of the median in the frequency distribution. The cumulative frequency for each interval is as follows:
- 2 – 8: 2
- 9 – 12: 2 + 7 = 9
- 13 – 15: 9 + 5 = 14
- 16 – 18: 14 + 8 = 22
- 19 – 22: 22 + 4 = 26
- 23 – 30: 26 + 4 = 30
Since there are 30 students, the median is at the ( 15^{th} ) value.
The cumulative frequency just above 15 is 22 at the interval 16 - 18. To interpolate:
Lower bound = 16 Cumulative frequency before this interval = 14 Frequency of this interval = 8
Using the formula: [ L + \left(\frac{\frac{n}{2} - CF}{f}\right) \times c]
Where:
- L = 16
- CF = 14
- f = 8
- c = width of the interval = 2
The calculation: [ 16 + \left(\frac{15 - 14}{8}\right) \times 2 = 16 + 0.25\times 2 = 16.5.]
So, the estimated median time taken is approximately 16.5 minutes.
Step 3
Estimate the mean and standard deviation of the times taken by the students.
Answer
To find the mean ( \bar{x} ) and standard deviation ( \sigma ):
-
Calculate the mean: [ \bar{x} = \frac{\sum f x}{\sum f} = \frac{477.5}{30} \approx 15.92
] -
Calculate the variance: To find the variance, we need to calculate ( \sum f x^2 ):
- Values of ( x^2 ): corresponding to midpoints 5, 10.5, 14, 17, 20.5, 26.5.
- Frequency multiplied by these midpoints to get ( fx^2 ).
- Σfx² = 8603.75 (given).
Variance ( \sigma^2 ) is calculated as follows: [ \sigma^2 = \frac{\sum f x^2}{\sum f} - \bar{x}^2 = \frac{8603.75}{30} - (15.92)^2 \approx 5.78
]
Thus, the mean is approximately 15.92 and the standard deviation is ( \sigma \approx \sqrt{5.78} \approx 2.4. )
Step 4
Give a reason to support the use of a normal distribution in this case.
Answer
A normal distribution can be used if the data is symmetrically distributed about the mean. Since the mean and median are similar, this suggests that the data may be closely approximated by a normal distribution. Additionally, if the range of data isn't too wide and the frequency distribution resembles a bell shape, normal distribution is a suitable assumption.
Step 5
Describe, giving a reason, the skewness of the times taken by the students to complete a different sudoku.
Answer
The skewness can be evaluated by comparing the quartiles. Given: [ Q_1 = 8.5, Q_2 = 13.0, Q_3 = 21.0. ]
To analyze skewness, calculate:
[
Q_3 - Q_2 = 21.0 - 13 = 8.0
]
[
Q_2 - Q_1 = 13.0 - 8.5 = 4.5
]
Since ( Q_3 - Q_2 > Q_2 - Q_1 ), the data is positively skewed or right-skewed, suggesting a longer tail on the right side of the distribution. This indicates that a few students took significantly longer to complete the sudoku.