$$ \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

Question

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

UpStudy AI · Accepted Answer

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!
There's a discrepancy in the given standard deviation $$ \sigma $$. Based on the calculations, $$ \sigma $$ should be approximately 0.102, not 7.6. Additionally, there are arithmetic errors in the third and fourth equations. The mean $$ M $$ should be between 101 and 111 to satisfy the inequality $$ M - 36 < 75 < M - 26 $$.

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Given:

Your Calculations:

Summary of Corrections:

Final Thoughts:

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