The mean and s.d. of 63 children on an arithmetic test are repectively `27*6` and `7*1` To them are added a new group of 26 who had less training and whose mean is `19*2` and s.d. `6*2` The values of the combined group differ from the original as to (i) the mean and (ii) the s.d.

Options

A. `2*5,0*7`
B. `2*3,0*8`
C. `1*5,0*9`
D. none of these

Answer

To solve the problem, we need to find the mean and standard deviation of the combined group of children after adding the new group. We will follow these steps:

### Step 1: Calculate the total score of the first group
The mean of the first group of 63 children is given as \( \bar{x_1} = 27.6 \).
To find the total score of this group, we can use the formula:
\[ \text{Total score of group 1} = \bar{x_1} \times n_1 = 27.6 \times 63 \]
Calculating this gives:
\[ \text{Total score of group 1} = 1742.8 \]

### Step 2: Calculate the total score of the second group
The mean of the second group of 26 children is given as \( \bar{x_2} = 19.2 \).
Using the same formula:
\[ \text{Total score of group 2} = \bar{x_2} \times n_2 = 19.2 \times 26 \]
Calculating this gives:
\[ \text{Total score of group 2} = 499.2 \]

### Step 3: Calculate the total score of the combined group
Now, we can find the total score of the combined group:
\[ \text{Total score of combined group} = \text{Total score of group 1} + \text{Total score of group 2} = 1742.8 + 499.2 = 2242 \]

### Step 4: Calculate the mean of the combined group
The total number of children in the combined group is:
\[ n = n_1 + n_2 = 63 + 26 = 89 \]
Now, we can find the mean of the combined group:
\[ \bar{x} = \frac{\text{Total score of combined group}}{n} = \frac{2242}{89} \approx 25.146 \]

### Step 5: Calculate the variance of the first group
The standard deviation of the first group is given as \( s_1 = 7.1 \). The variance is:
\[ \sigma_1^2 = s_1^2 = (7.1)^2 = 50.41 \]

### Step 6: Calculate the variance of the second group
The standard deviation of the second group is given as \( s_2 = 6.2 \). The variance is:
\[ \sigma_2^2 = s_2^2 = (6.2)^2 = 38.44 \]

### Step 7: Calculate the combined variance
The formula for the combined variance \( \sigma^2 \) is:
\[ \sigma^2 = \frac{(n_1 - 1) \sigma_1^2 + (n_2 - 1) \sigma_2^2 + n_1 (\bar{x_1} - \bar{x})^2 + n_2 (\bar{x_2} - \bar{x})^2}{n - 1} \]
Substituting the values:
\[ \sigma^2 = \frac{(63 - 1) \cdot 50.41 + (26 - 1) \cdot 38.44 + 63 \cdot (27.6 - 25.146)^2 + 26 \cdot (19.2 - 25.146)^2}{89 - 1} \]
Calculating each term:
- \( (63 - 1) \cdot 50.41 = 62 \cdot 50.41 = 3125.42 \)
- \( (26 - 1) \cdot 38.44 = 25 \cdot 38.44 = 961 \)
- \( 63 \cdot (27.6 - 25.146)^2 = 63 \cdot (2.454)^2 \approx 63 \cdot 6.034 = 379.142 \)
- \( 26 \cdot (19.2 - 25.146)^2 = 26 \cdot (-5.946)^2 \approx 26 \cdot 35.353 = 919.178 \)

Now summing these:
\[ \sigma^2 = \frac{3125.42 + 961 + 379.142 + 919.178}{88} \approx \frac{5384.74}{88} \approx 61.4 \]

### Step 8: Calculate the standard deviation of the combined group
The standard deviation is the square root of the variance:
\[ \sigma = \sqrt{61.4} \approx 7.84 \]

### Step 9: Calculate the differences in mean and standard deviation
Now we can find the differences:
- Difference in mean:
\[ \text{Difference in mean} = \bar{x} - \bar{x_1} = 25.146 - 27.6 \approx -2.454 \]
- Difference in standard deviation:
\[ \text{Difference in standard deviation} = \sigma - s_1 = 7.84 - 7.1 \approx 0.74 \]

### Final Results
- The difference in mean is approximately \( -2.454 \) (which means the mean decreased).
- The difference in standard deviation is approximately \( 0.74 \) (which means the standard deviation increased).