Scores on a test are normally distributed with a mean of 70 and standard deviation of 10 . Applying the Empirical Rule, we would expect the middle $$ 95 \% $$ of scores to fall between what two values? 40 and 100

Question

UpStudy AI · Accepted Answer

The **Empirical Rule**, also known as the **68-95-99.7 Rule**, is a guideline that applies to normally distributed data. It helps estimate the percentage of data points that fall within certain ranges of the mean. Here's a breakdown:

1. **Approximately 68%** of the data falls **within 1 standard deviation** of the mean.
2. **Approximately 95%** of the data falls **within 2 standard deviations** of the mean.
3. **Approximately 99.7%** of the data falls **within 3 standard deviations** of the mean.

Given your scenario:

- **Mean (μ)** = 70
- **Standard Deviation (σ)** = 10

### Applying the Empirical Rule:

1. **Within 1 Standard Deviation (68%)**:
   - Lower bound: μ - σ = 70 - 10 = **60**
   - Upper bound: μ + σ = 70 + 10 = **80**
   - **Range**: 60 to 80

2. **Within 2 Standard Deviations (95%)**:
   - Lower bound: μ - 2σ = 70 - 20 = **50**
   - Upper bound: μ + 2σ = 70 + 20 = **90**
   - **Range**: 50 to 90

3. **Within 3 Standard Deviations (99.7%)**:
   - Lower bound: μ - 3σ = 70 - 30 = **40**
   - Upper bound: μ + 3σ = 70 + 30 = **100**
   - **Range**: 40 to 100

### Addressing Your Statement:

You mentioned that the middle **95%** of scores fall between **40 and 100**. However, according to the Empirical Rule:

- **95%** of the data falls **between 50 and 90** (±2 standard deviations).
- The range **40 to 100** actually encompasses **approximately 99.7%** of the data (±3 standard deviations).

### **Conclusion:**

For the **middle 95%** of test scores:

- **Expected Range**: **50 to 90**

The range **40 to 100** would cover a much larger portion (**~99.7%**) of the data.
The middle 95% of test scores fall between 50 and 90.

Scores on a test are normally distributed with a mean of 70 and standard deviation of 10 . Applying the Empirical Rule, we would expect the middle \( 95 \% \) of scores to fall between what two values? 40 and 100

Real Tutor Solution

Answer

Solution

Extra Insights

Related Questions

Latest Statistics Questions