1. A has an arithmetic mean of 50 and standard deviation of 5 . B has an arithmetic mean of 48 . What standard deviation will \( B \) hare gor both of their arithmetic mean to be representative

To determine the standard deviation that \( B \) should have for both \( A \) and \( B \) to have representative arithmetic means, we need to consider the concept of the combined mean and how standard deviations relate to the means. Given: - \( A \) has a mean (\( \mu_A \)) of 50 and a standard deviation (\( \sigma_A \)) of 5. - \( B \) has a mean (\( \mu_B \)) of 48. To find a standard deviation for \( B \) that makes the means representative, we can consider the following: 1. **Combined Mean**: If we want the means to be representative, we can look at the combined mean of both groups. The combined mean (\( \mu_{combined} \)) can be calculated as: \[ \mu_{combined} = \frac{\mu_A + \mu_B}{2} = \frac{50 + 48}{2} = 49 \] 2. **Standard Deviation**: The standard deviation of the combined means depends on the individual standard deviations and the number of observations in each group. However, without knowing the sample sizes of \( A \) and \( B \), we cannot calculate the exact standard deviation for \( B \) that would make the means representative. If we assume that both groups have the same number of observations (let's say \( n \)), the combined standard deviation (\( \sigma_{combined} \)) can be calculated using the formula for the pooled standard deviation: \[ \sigma_{combined} = \sqrt{\frac{(n-1)\sigma_A^2 + (n-1)\sigma_B^2}{2n - 2}} \] However, without additional information about the sample sizes or the desired relationship between the two standard deviations, we cannot determine a specific value for \( \sigma_B \). In summary, to find a standard deviation for \( B \) that makes the means representative, we need more information about the sample sizes or the context in which we want the means to be representative. If you have specific criteria or additional data, please provide that for a more precise answer.