Suppose that mean retail price per gallon of regular grade gasoline is $3.42 with a standard deviation of $0.10 and that the retail price per gallon has a bell-shaped distribution.  What percentage of regular grade gasoline sells for between $3.22 and $3.62 per gallon

To determine the percentage of regular grade gasoline that sells for between $3.22 and $3.62 per gallon, we can use the Empirical Rule (also known as the 68-95-99.7 Rule) which applies to bell-shaped (normal) distributions.

### Given:
- **Mean (μ):** $3.42
- **Standard deviation (σ):** $0.10
- **Range:** $3.22 to $3.62

### Steps:

1. **Calculate the z-scores** for the boundaries of $3.22 and $3.62.

   \[
   z_1 = \frac{3.22 - 3.42}{0.10} = \frac{-0.20}{0.10} = -2
   \]
   \[
   z_2 = \frac{3.62 - 3.42}{0.10} = \frac{0.20}{0.10} = 2
   \]

2. **Interpret the z-scores:**
   - A z-score of -2 corresponds to 2 standard deviations below the mean.
   - A z-score of 2 corresponds to 2 standard deviations above the mean.

3. **Apply the Empirical Rule:**
   - According to the rule, approximately **95%** of the data in a normal distribution falls within 2 standard deviations of the mean.

Therefore, **95%** of regular grade gasoline sells for between $3.22 and $3.62 per gallon.

Would you like more details or have any questions?

### Related Questions:
1. What is the percentage of gasoline prices that fall between $3.32 and $3.52?
2. How much of the gasoline is expected to be priced above $3.62 per gallon?
3. What is the price range that captures 68% of the gasoline prices?
4. How would the percentage change if the standard deviation were $0.20 instead of $0.10?
5. Can you determine the percentage of gasoline prices below $3.22?

### Tip:
When working with normal distributions, understanding how to calculate and interpret z-scores is key to finding probabilities and percentages for specific ranges.

Learn how to calculate the percentage of regular grade gasoline prices that fall between $3.22 and $3.62 per gallon using the Empirical Rule for normal distributions. Given a mean of $3.42 and a standard deviation of $0.10, understand how to interpret z-scores and apply statistical rules to solve this problem.

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