urrent Attempt in Progress Use the $$ 95 \% $$ rule and the fact that the summary statistics come from a distribution that is symmetric and bell-shaped to find an interval that is expected to contain about $$ 95 \% $$ of the data values.

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To determine an interval that is expected to contain approximately $$ 95\% $$ of the data values for a symmetric and bell-shaped (normally distributed) dataset, you can use the **68-95-99.7 (Empirical) Rule**. Specifically, the **95% rule** states that about $$ 95\% $$ of the data lies within two standard deviations of the mean.

### Step-by-Step Guide:

1. **Identify the Summary Statistics:**
   - **Mean ($$ \mu $$)**: The average of all data points.
   - **Standard Deviation ($$ \sigma $$)**: Measures the dispersion or variability of the data points from the mean.

2. **Apply the 95% Rule:**
   - Calculate the interval by adding and subtracting **two standard deviations** from the mean.
   - **Interval Formula:**
     $$
     \text{Lower Bound} = \mu - 2\sigma
     $$
     $$
     \text{Upper Bound} = \mu + 2\sigma
     $$

3. **Construct the Interval:**
   - **Interval:** $$ [\mu - 2\sigma, \mu + 2\sigma] $$

### Example:

Suppose you have the following summary statistics for a dataset:

- **Mean ($$ \mu $$)**: 50
- **Standard Deviation ($$ \sigma $$)**: 5

**Applying the 95% Rule:**

$$
\text{Lower Bound} = 50 - 2(5) = 50 - 10 = 40
$$
$$
\text{Upper Bound} = 50 + 2(5) = 50 + 10 = 60
$$

**Therefore, the interval $$[40, 60]$$ is expected to contain about $$ 95\% $$ of the data values.**

### Visual Representation:

```
Mean (μ) = 50
σ = 5

Interval: [50 - 10, 50 + 10] = [40, 60]

|-----|-----|-----|-----|-----|
40    45    50    55    60
```

### Important Notes:

- **Assumption of Normality:** The 95% rule is most accurate for data that follows a **normal distribution**. For distributions that are symmetric and bell-shaped but not perfectly normal, the approximation still holds reasonably well.
  
- **Outliers and Skewness:** If the data has significant outliers or is skewed, the 95% interval based on mean and standard deviation may not accurately capture the middle 95% of the data.

- **Use of Multipliers:** While **2** is commonly used for a rough estimate, the exact multiplier for a 95% interval in a normal distribution is approximately **1.96**. However, using 2 simplifies calculations without significantly compromising accuracy for most practical purposes.

### Conclusion

By utilizing the mean and standard deviation of your dataset along with the 95% rule, you can confidently construct an interval that is expected to contain about $$ 95\% $$ of your data values, provided the data distribution is symmetric and bell-shaped.
To find an interval containing about 95% of the data, subtract and add two standard deviations from the mean. So, the interval is from (mean - 2σ) to (mean + 2σ).

urrent Attempt in Progress Use the \( 95 \% \) rule and the fact that the summary statistics come from a distribution that is symmetric and bell-shaped to find an interval that is expected to contain about \( 95 \% \) of the data values.

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