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Luke is an office receptionist - OCR - GCSE Maths - Question 20 - 2019 - Paper 1
Question 20
Luke is an office receptionist. Each day, for 60 days, he records the number of people visiting the office. Number of people, (n) Frequency 0 < n ≤ 5 ... show full transcript
Worked Solution & Example Answer:Luke is an office receptionist - OCR - GCSE Maths - Question 20 - 2019 - Paper 1
Step 1
Calculate an estimate of the mean number of people visiting the office.
Answer
To estimate the mean number of people visiting the office, we first need to find the midpoints of the given intervals:
- For the interval (0, 5], the midpoint is ( m_1 = \frac{0 + 5}{2} = 2.5 )
- For the interval (5, 10], the midpoint is ( m_2 = \frac{5 + 10}{2} = 7.5 )
- For the interval (10, 20], the midpoint is ( m_3 = \frac{10 + 20}{2} = 15 )
- For the interval (20, 40], the midpoint is ( m_4 = \frac{20 + 40}{2} = 30 )
Next, we calculate the weighted sum of midpoints multiplied by their frequencies:
[ \text{Estimated Mean} = \frac{(m_1 \times f_1) + (m_2 \times f_2) + (m_3 \times f_3) + (m_4 \times f_4)}{\text{Total Frequency}} ]
Substituting the values:
[ \text{Estimated Mean} = \frac{(2.5 \times 20) + (7.5 \times 14) + (15 \times 11) + (30 \times 15)}{60} ] [ = \frac{50 + 105 + 165 + 450}{60} = \frac{770}{60} \approx 12.83 ] Thus, the estimated mean number of people visiting the office is approximately 12.83.
Step 2
Explain why he may be wrong.
Answer
Luke states that the range is 40. However, to determine the range, we need to find the minimum and maximum values in the data:
- The minimum number of people visiting is from the interval (0, 5], which is 0.
- The maximum number of people is from the interval (20, 40], with a maximum of 40.
Thus, the range would be calculated as:
[ \text{Range} = \text{Maximum} - \text{Minimum} = 40 - 0 = 40 ]
While this shows that Luke's reported range is mathematically correct based on the given intervals, he may be misinterpreting the data if he believes that it could not account for more precise details, such as the actual upper limit of the range exceeding the maximum interval noted (like 40), which could lead him to conclude that the range could instead be wider than stated.