Observe the following statements :
(A) : 10 is the mean of a set of 7 obervations and 5 is the mean of a set of 3 observations. The mean of a combined set is 9.
(R ) : If `bar(x)_(i)(i = 1,2,…,k)` are the means of k - series `n_(i)(I = 1,2,3,…,k)` respectively, then the combined or composite mean is
`bar(x) = (n_(1)bar(x)_(1) + n_(2) bar(x)_(2) + ...+ n_(k)bar(x_(k)))/(n_(1)+n_(2)+...+n_(k))`

Options

A. both A, R are true and `R rArr A`
B. both A, R are true and `R cancelrArr A`
C. A is true, R is false
D. A is false, R is true

Answer

To solve the problem, we need to analyze the two statements provided:

**Assertion (A)**: The mean of a set of 7 observations is 10, and the mean of a set of 3 observations is 5. The mean of the combined set is 9.

**Reason (R)**: The formula for the combined mean of k series is given by:
\[ \bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2 + \ldots + n_k \bar{x}_k}{n_1 + n_2 + \ldots + n_k} \]

### Step-by-Step Solution:

1. **Identify the values**:
- For the first set of observations:
- Number of observations, \( n_1 = 7 \)
- Mean, \( \bar{x}_1 = 10 \)
- For the second set of observations:
- Number of observations, \( n_2 = 3 \)
- Mean, \( \bar{x}_2 = 5 \)

2. **Calculate the total sums of the observations**:
- Total sum of the first set:
\[ S_1 = n_1 \cdot \bar{x}_1 = 7 \cdot 10 = 70 \]
- Total sum of the second set:
\[ S_2 = n_2 \cdot \bar{x}_2 = 3 \cdot 5 = 15 \]

3. **Calculate the combined sum of the observations**:
- Combined sum:
\[ S = S_1 + S_2 = 70 + 15 = 85 \]

4. **Calculate the total number of observations**:
- Total number of observations:
\[ N = n_1 + n_2 = 7 + 3 = 10 \]

5. **Calculate the combined mean**:
- Using the formula for the combined mean:
\[ \bar{x} = \frac{S}{N} = \frac{85}{10} = 8.5 \]

6. **Compare the calculated mean with the assertion**:
- The calculated mean is 8.5, which is not equal to 9 as stated in the assertion.

### Conclusion:
- **Assertion (A)** is false because the combined mean is 8.5, not 9.
- **Reason (R)** is true as it correctly states the formula for calculating the combined mean.

### Final Answer:
- The correct conclusion is: Assertion (A) is false, and Reason (R) is true.

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