Intermediate November 2023 Unit 2 Q21
21. The grouped frequency table below shows data for 32 values of \(t\).
One of the frequencies in the table is represented by \(a\).
| \(t\) | Frequency |
|---|---|
| \(20 \leqslant t \lt 30\) | 2 |
| \(30 \leqslant t \lt 40\) | 8 |
| \(40 \leqslant t \lt 50\) | 4 |
| \(50 \leqslant t \lt 60\) | \(a\) |
| \(60 \leqslant t \lt 70\) | 3 |
| \(70 \leqslant t \lt 80\) | 5 |
Find the estimated mean value of \(t\) for the 32 values. [5]
| Scheme | Marks | Comments |
|---|---|---|
| Midpoints 25, 35, 45, (55), 65, 75 | B1 | May be implied in later working (i.e the correct products). |
| Missing 10 for \(50 \leqslant t \lt 60\) | B1 | |
| 25×2 + 35×8 + 45×4 + 55×10 + 65×3 + 75×5 (= 50 + 280 + 180 + 550 + 195 + 375 = 1630) | M1 | FT ‘their 10’ provided ≠ 0 or 1. Allow with consistent incorrect midpoints provided at least 5 within the correct interval including ‘bounds’ Allow use of \(a\) instead of 10 (sight of 1080 + 55a). |
| ÷ 32 | m1 | FT 22 + ‘their \(a\)’ (a ≠ 0). Allow use of \(a\) instead of 10. |
| 50·9(375) or 51 or equivalent ISW | A1 | CAO. Must be derived from correct working. If no marks or first B1 only, award SC1 for one of the following:
\(\dfrac{1080 + 55a}{22 + a}\) or \(\dfrac{1080 + 55a}{32}\) or equivalent expression involving \(a\). |