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Each of 60 students was asked to draw a 20° angle without using a protractor - Edexcel - A-Level Maths Statistics - Question 1 - 2015 - Paper 1
Question 1
Each of 60 students was asked to draw a 20° angle without using a protractor. The size of each angle drawn was measured. The results are summarised in the box plot b... show full transcript
Worked Solution & Example Answer:Each of 60 students was asked to draw a 20° angle without using a protractor - Edexcel - A-Level Maths Statistics - Question 1 - 2015 - Paper 1
Step 1
Step 2
Step 3
Use linear interpolation to estimate the size of the median angle drawn. Give your answer to 1 decimal place.
Answer
To find the median for the 70° angles, we need the combined cumulative frequency:
- 55 ≤ a < 60: 6 students
- 60 ≤ a < 65: 15 students (Cumulative: 21)
- 65 ≤ a < 70: 11 students (Cumulative: 32)
- 70 ≤ a < 75: 8 students (Cumulative: 40)
- 75 ≤ a < 80: 7 students (Cumulative: 47)
- 80 ≤ a < 85: 7 students (Cumulative: 54)
The median position is at (60/2) = 30. The median is found by interpolating between the 30th and 31st values: Since 32 students are less than 70°, we find that the median angle is approximately:
Median = 65 + rac{(30 - 21)}{(32 - 21)} imes (70 - 65) = 65 + 4.5 = 69.5°Thus, the estimated median angle is approximately 69.5°.
Step 4
Show that the lower quartile is 63°.
Answer
To verify the lower quartile (Q1), we look at the cumulative frequencies:
- The first 15 (Q1 position, which corresponds to the 15th student) lie between 60-65°.
- Continuing to 6 more (Cumulative: 21), we count further up to find that the angle at Q1 is 63° since the lower 25% of 60 is 15 students and their median is between 60 and 65 degrees.
Thus, it's shown that Q1 = 63°.
Step 5
Show that there are no outliers for these data.
Answer
To determine outliers:
- Calculate 1.5 * IQR = 1.5 * 13 = 19.5
- Calculate the lower limit = Q1 - 19.5 = 63 - 19.5 = 43.5
- Calculate the upper limit = Q3 + 19.5 = 75 + 19.5 = 94.5
We can see all angles (55 to 84) lie within [43.5, 94.5]. Therefore, there are no outliers in these data.
Step 6
Draw a box plot for these data on the grid on page 3.
Answer
To draw the box plot:
- The minimum value is 55°, the lower quartile (Q1) is 63°, the median is 69.5°, the upper quartile (Q3) is 75°, and the maximum is 84°. Plot these points on the grid and connect Q1 and Q3 with a box, marking the median within this box, and extending lines to the minimum and maximum to complete the plot.
Step 7
State which angle the students were more accurate at drawing. Give reasons for your answer.
Answer
The median of the 70° angles (approximately 69.5°) is closer to 70° compared to the median of the 20° angles (which is 20°). Therefore, the students were more accurate at drawing the 70° angle as the discrepancies are smaller, and the IQR for 70° is also less than for 20°. This indicates fewer variations in their responses.