The mean of a binomial distribution is 15 and standard deviation is 5, then which one of the following is correct ?

Options

A. `p=2/3`
B. `q=5/3`
C. data's are absolutely correct
D. data's are absolutely wrong

Answer

To solve the problem, we need to analyze the given information about the binomial distribution, specifically the mean and standard deviation.

### Step-by-Step Solution:

1. **Understanding the Mean and Standard Deviation of a Binomial Distribution:**
- The mean (μ) of a binomial distribution is given by the formula:
\[ \mu = n \cdot p \]
- The standard deviation (σ) is given by:
\[ \sigma = \sqrt{n \cdot p \cdot q} \]
where \( q = 1 - p \).

2. **Given Values:**
- Mean (μ) = 15
- Standard Deviation (σ) = 5

3. **Calculating Variance:**
- The variance (σ²) is the square of the standard deviation:
\[ \sigma^2 = 5^2 = 25 \]

4. **Setting Up the Equations:**
- From the mean:
\[ n \cdot p = 15 \quad \text{(1)} \]
- From the variance:
\[ n \cdot p \cdot q = 25 \quad \text{(2)} \]
- Since \( q = 1 - p \), we can substitute \( q \) in equation (2):
\[ n \cdot p \cdot (1 - p) = 25 \quad \text{(3)} \]

5. **Substituting from Equation (1) into Equation (3):**
- From equation (1), we can express \( n \) in terms of \( p \):
\[ n = \frac{15}{p} \]
- Substitute \( n \) into equation (3):
\[ \frac{15}{p} \cdot p \cdot (1 - p) = 25 \]
- Simplifying this gives:
\[ 15(1 - p) = 25 \]
- Rearranging:
\[ 15 - 15p = 25 \]
\[ -15p = 25 - 15 \]
\[ -15p = 10 \]
\[ p = -\frac{10}{15} = -\frac{2}{3} \]

6. **Analyzing the Result:**
- The value of \( p \) must be between 0 and 1 for a binomial distribution. Since \( p = -\frac{2}{3} \) is not a valid probability, this indicates that the data provided in the question is incorrect.

7. **Conclusion:**
- Therefore, the correct answer is that the data is absolutely wrong.

### Final Answer:
The data is absolutely wrong (Option 4).