The means and standard devation of the marks of `200` candidates in half early exam were found to be `40` and `15` respectively. The marks of the candidates exceeds their marks by `10` in final exam. The mean and standard deviation of the marks in final exam are respectively ?

Options

A. `50, 25`
B. `50, 15`
C. `30, 25`
D. `30, 15`

Explanation

To solve the problem step by step, we will follow the information given in the question and apply the relevant statistical concepts.

### Step 1: Identify the given data
- Number of candidates (n) = 200
- Mean of half-yearly exam (μ₁) = 40
- Standard deviation of half-yearly exam (σ₁) = 15
- Increase in marks for final exam = 10

### Step 2: Calculate the mean of the final exam
The mean of the final exam can be calculated by adding the increase in marks to the mean of the half-yearly exam.

\[ \text{Mean of final exam} (\mu_2) = \text{Mean of half-yearly exam} + \text{Increase in marks} \]
\[ \mu_2 = 40 + 10 = 50 \]

### Step 3: Calculate the standard deviation of the final exam
The standard deviation is not affected by a change in origin (i.e., adding or subtracting a constant). Therefore, the standard deviation of the final exam remains the same as that of the half-yearly exam.

\[ \text{Standard deviation of final exam} (\sigma_2) = \text{Standard deviation of half-yearly exam} = 15 \]

### Step 4: State the final results
- Mean of the final exam = 50
- Standard deviation of the final exam = 15

### Conclusion
The mean and standard deviation of the marks in the final exam are respectively **50** and **15**.

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