5. Mean $$ =95 ; $$ Standard Deviation $$ =12 $$

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It looks like you've provided the **mean** and **standard deviation** for a dataset or a probability distribution:

- **Mean ($$ \mu $$) = 95**
- **Standard Deviation ($$ \sigma $$) = 12**

These two statistics are fundamental in understanding the distribution of your data. Here's a breakdown of what they represent and how you might use them:

### 1. Understanding Mean and Standard Deviation

- **Mean ($$ \mu $$)**: This is the average value of your dataset. It provides a central value around which the data points are distributed.
  
- **Standard Deviation ($$ \sigma $$)**: This measures the amount of variation or dispersion in your dataset. A smaller standard deviation indicates that the data points are close to the mean, while a larger one suggests more spread out data.

### 2. Assuming a Normal Distribution

If your data follows a **normal distribution** (a common assumption in statistics), you can use the mean and standard deviation to make various probabilistic statements about the data.

#### a. **Empirical Rule (68-95-99.7 Rule)**

- **68%** of the data falls within **±1 standard deviation** from the mean:
  - $$ 95 - 12 = 83 $$ to $$ 95 + 12 = 107 $$
  
- **95%** of the data falls within **±2 standard deviations**:
  - $$ 95 - 24 = 71 $$ to $$ 95 + 24 = 119 $$
  
- **99.7%** of the data falls within **±3 standard deviations**:
  - $$ 95 - 36 = 59 $$ to $$ 95 + 36 = 131 $$

#### b. **Calculating Z-Scores**

A **z-score** indicates how many standard deviations an element is from the mean. It's calculated as:

$$
z = \frac{(X - \mu)}{\sigma}
$$

Where:
- $$ X $$ = value from the dataset
- $$ \mu $$ = mean
- $$ \sigma $$ = standard deviation

**Example:**
If you want to find how unusual a score of **110** is:

$$
z = \frac{(110 - 95)}{12} = \frac{15}{12} = 1.25
$$

A z-score of **1.25** means that **110** is **1.25** standard deviations above the mean.

#### c. **Finding Probabilities**

Using z-scores and standard normal distribution tables (or software), you can find the probability of a value occurring below or above a certain point.

**Example:**
What's the probability of selecting a value greater than **107**?

1. Calculate the z-score for **107**:
   $$
   z = \frac{(107 - 95)}{12} = 1
   $$
   
2. Look up the z-score of **1** in a standard normal distribution table, which gives the probability of a value being less than **107** (approximately **84.13%**).

3. To find the probability of a value being greater than **107**:
   $$
   1 - 0.8413 = 0.1587 \text{ or } 15.87\%
   $$

### 3. Applications

Depending on your specific needs, here are some ways you might use the mean and standard deviation:

- **Assessing Performance**: Determine how individual data points compare to the average.
- **Quality Control**: Monitor processes to ensure consistency.
- **Research and Analysis**: Understand the distribution and variability in your data.

### 4. Additional Considerations

- **Distribution Shape**: While the normal distribution is common, not all datasets follow it. It's essential to visualize your data (e.g., using histograms) to understand its distribution.
  
- **Outliers**: Extreme values can affect the mean and standard deviation. Consider detecting and handling outliers appropriately.

### 5. Example Problem

**Problem:**  
Assume that the scores of a test are normally distributed with a mean of **95** and a standard deviation of **12**.

a. **What percentage of students scored between 83 and 107?**

**Solution:**  
This range is within **±1 standard deviation** from the mean.

- According to the empirical rule, approximately **68%** of the data lies within this range.

b. **What is the probability that a randomly selected student scored above 119?**

**Solution:**

1. Calculate the z-score for **119**:
   $$
   z = \frac{(119 - 95)}{12} = \frac{24}{12} = 2
   $$
   
2. The probability of a z-score being less than **2** is approximately **97.5%**.

3. Therefore, the probability of scoring above **119** is:
   $$
   100\% - 97.5\% = 2.5\%
   $$

---

If you have a specific question or need assistance with a particular problem related to these statistics, feel free to provide more details!
The mean is 95 and the standard deviation is 12.

5. Mean \( =95 ; \) Standard Deviation \( =12 \)

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