The number of patients per shift ( X) consulting Dr. Gita in her past 100 shifts is
shown in the figure. If the amount she earns is ₹ 1000(X - 0.2), what is the average
amount (in ₹) she has earned per shift in the past 100 shifts?
Note: The figure shown is representative.
Options:
(A) 6,100
(B) 6,300
(C) 6,000
(D) 6,500
20
40
30
10
0
10
20
30
40
50
5 6 7 8
Number of shifts
Number of patients per shift (X)
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.10)**
**1. Intuitive Physical Setup & Given Parameters**:
A bar chart shows the distribution of the number of patients $X$ per shift seen by Dr. Gita over $N = 100$ shifts:
- $X = 5$ patients: 20 shifts
- $X = 6$ patients: 40 shifts
- $X = 7$ patients: 30 shifts
- $X = 8$ patients: 10 shifts
- Total shifts: $20 + 40 + 30 + 10 = 100$
Earning formula per shift:
$\text{Earning}(X) = ₹\,1000(X - 0.2)$
We need to find her **average earning per shift** over the past 100 shifts.
**2. Step-by-Step Statistical Derivation**:
1. **Compute Average Number of Patients per Shift ($\bar{X}$)**:
$\bar{X} = \frac{\sum (X_i \times f_i)}{N} = \frac{(5 \times 20) + (6 \times 40) + (7 \times 30) + (8 \times 10)}{100}$
$\sum (X_i f_i) = 100 + 240 + 210 + 80 = 630\text{ patients}$
$\bar{X} = \frac{630}{100} = 6.3\text{ patients/shift}$
2. **Compute Average Earning**:
Using the linearity of the expectation operator $E[aX + b] = a E[X] + b$:
$\text{Average Earning} = 1000(\bar{X} - 0.2)$
$\text{Average Earning} = 1000(6.3 - 0.2) = 1000(6.1) = ₹\,6,100$
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 6,100**: ✅ **CORRECT**. $1000(6.3 - 0.2) = ₹\,6,100$.
- **(B) 6,300**: ❌ **INCORRECT**. Forgets the subtraction of $0.2$ ($1000 \times 6.3$).
- **(C) 6,000**: ❌ **INCORRECT**. Rounding error.
- **(D) 6,500**: ❌ **INCORRECT**. Unweighted arithmetic average of $\{5, 6, 7, 8\} = 6.5$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: By linearity of expectation, calculate the weighted mean $\bar{X} = 6.3$ first, then evaluate the linear cost function $E[1000(X - 0.2)] = 1000(6.3 - 0.2) = 6100$.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.