\( \left. \begin{array} { l }{ ( \frac { ( - 1 ) ^ { 2 } } { \sigma } ) ^ { 2 } = 95 \quad \sigma = 7.6 \quad x = 75 } \\ { M - \sigma = 95 - 7.6 = 87.4 } \\ { M - 26 = 95 - 2 ( 7.6 ) = 95 - 15.2 = 77.8 } \\ { M - 36 = 95 - 3 ( 2.6 ) = 95 - 22.8 = 72 : 2 } \\ { M - 36 < 75 < M - 26 } \end{array} \right. \)

It looks like you're working through a problem involving the mean \( M \), standard deviation \( \sigma \), and a data point \( x = 75 \). Let's break down each of your steps to ensure accuracy and clarity. ### Given: - \( \sigma = 7.6 \) - \( x = 75 \) ### Your Calculations: 1. **First Equation:** \[ \left( \frac{(-1)^2}{\sigma} \right)^2 = 95 \] **Simplification:** \[ \left( \frac{1}{\sigma} \right)^2 = 95 \quad \Rightarrow \quad \frac{1}{\sigma^2} = 95 \quad \Rightarrow \quad \sigma^2 = \frac{1}{95} \quad \Rightarrow \quad \sigma \approx \frac{1}{9.75} \approx 0.102 \] **Issue Identified:** - **Conflict:** You've stated \( \sigma = 7.6 \) but the equation suggests \( \sigma \approx 0.102 \). There's a discrepancy here. Ensure that the initial equation correctly represents your problem. 2. **Second Equation:** \[ M - \sigma = 95 - 7.6 = 87.4 \] **Solving for \( M \):** \[ M = 87.4 + \sigma = 87.4 + 7.6 = 95 \] **Result:** \[ M = 95 \] 3. **Third Equation:** \[ M - 26 = 95 - 2 \times 7.6 = 95 - 15.2 = 79.8 \quad (\text{Note: You wrote } 77.8 \text{, which seems to be a miscalculation}) \] **Correct Calculation:** \[ 95 - 15.2 = 79.8 \] **Solving for \( M \):** \[ M = 79.8 + 26 = 105.8 \] 4. **Fourth Equation:** \[ M - 36 = 95 - 3 \times 2.6 = 95 - 7.8 = 87.2 \] **Note on Calculation:** - It appears you intended to use \( 3 \times \sigma \) (where \( \sigma = 7.6 \)), which should be \( 3 \times 7.6 = 22.8 \), not \( 3 \times 2.6 \). **Correct Calculation:** \[ M - 36 = 95 - 22.8 = 72.2 \quad \Rightarrow \quad M = 72.2 + 36 = 108.2 \] **You wrote:** \( 72 : 2 \), which is unclear. If you meant \( 72.2 \div 2 = 36.1 \), but it's not aligned with the equation. Likely, it's a typographical error. 5. **Inequality:** \[ M - 36 < 75 < M - 26 \] **Solving the Inequality:** - **Left Part:** \( M - 36 < 75 \quad \Rightarrow \quad M < 75 + 36 = 111 \) - **Right Part:** \( 75 < M - 26 \quad \Rightarrow \quad M > 75 + 26 = 101 \) **Combined Inequality:** \[ 101 < M < 111 \] **Interpretation:** The mean \( M \) must be between 101 and 111 to satisfy the condition that \( 75 \) lies between \( M - 36 \) and \( M - 26 \). ### Summary of Corrections: 1. **First Equation:** There's a mismatch between the given \( \sigma = 7.6 \) and the equation leading to \( \sigma \approx 0.102 \). Verify the original equation for accuracy. 2. **Third Equation:** Ensure accurate arithmetic when subtracting \( 2 \times \sigma \). 3. **Fourth Equation:** Confirm whether the multiplier is \( 2.6 \) or if it's a typo meant to be \( 7.6 \). Using \( \sigma = 7.6 \) aligns with standard deviations multiples. 4. **Final Inequality:** Your solution correctly identifies that \( M \) should be between 101 and 111 to satisfy \( M - 36 < 75 < M - 26 \). ### Final Thoughts: - **Consistency is Key:** Ensure that all equations and substitutions consistently use the correct values for variables like \( \sigma \). - **Double-Check Arithmetic:** Small calculation errors can lead to significant discrepancies in your results. - **Clarify the Problem Context:** Providing the context or the original problem statement can help in offering more precise guidance. If you could share the original problem or clarify the context further, I can assist you more effectively!