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 $3.52 per gallon

To determine the percentage of regular grade gasoline that sells for between $3.22 and $3.52 per gallon, we can again use the Empirical Rule and z-scores.

### Given:
- **Mean (μ):** $3.42
- **Standard deviation (σ):** $0.10
- **Range:** $3.22 to $3.52

### Steps:

1. **Calculate the z-scores** for the boundaries of $3.22 and $3.52.

   \[
   z_1 = \frac{3.22 - 3.42}{0.10} = \frac{-0.20}{0.10} = -2
   \]
   \[
   z_2 = \frac{3.52 - 3.42}{0.10} = \frac{0.10}{0.10} = 1
   \]

2. **Interpret the z-scores:**
   - A z-score of -2 corresponds to 2 standard deviations below the mean.
   - A z-score of 1 corresponds to 1 standard deviation above the mean.

3. **Apply the Empirical Rule:**
   - Approximately **68%** of the data in a normal distribution falls within 1 standard deviation of the mean (from $ z = -1 $ to $ z = 1 $).
   - Approximately **95%** of the data falls within 2 standard deviations of the mean (from $ z = -2 $ to $ z = 2 $).

To find the percentage between $ z = -2 $ and $ z = 1 $:
- From $ z = -2 $ to $ z = 2 $ covers 95% of the distribution.
- From $ z = -2 $ to $ z = -1 $ covers approximately 13.5% (half of the range from $ z = -2 $ to $ z = -1 $).

So, the percentage between $ z = -2 $ and $ z = 1 $ is:
- $ 68\% $ (from $ z = -1 $ to $ z = 1 $) + $ 13.5\% $ (from $ z = -2 $ to $ z = -1 $)

Therefore, approximately **81.5%** of regular grade gasoline sells for between $3.22 and $3.52 per gallon.

Would you like more details or have any questions?

### Related Questions:
1. What percentage of gasoline prices fall between $3.32 and $3.52?
2. How much of the gasoline is expected to be priced below $3.22 per gallon?
3. What is the percentage of gasoline prices between $3.42 and $3.62?
4. How would the percentage change if the range was $3.32 to $3.62?
5. What is the probability that a randomly selected gallon of gasoline costs more than $3.52?

### Tip:
When using the Empirical Rule, breaking down the ranges into segments of standard deviations helps to determine the cumulative percentage accurately.

Calculate the percentage of regular grade gasoline prices that fall between $3.22 and $3.52 per gallon using the normal distribution, z-scores, and the Empirical Rule. Given mean retail price per gallon of $3.42 and standard deviation of $0.10, understand how to apply statistical concepts to analyze price ranges.

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