What is the approximate percentage of buyers who paid less than $1440?
%

What is the approximate percentage of buyers who paid between $1520 and $1680?
%

What is the approximate percentage of buyers who paid less than $1360?
%

What is the approximate percentage of buyers who paid between $1600 and $1760?

To answer these questions, we will use the properties of the normal distribution and the empirical rule (68-95-99.7 rule), which provides approximate percentages within certain standard deviations from the mean.

### Given Information:
- **Mean ($\mu$)**: $1600
- **Standard Deviation ($\sigma$)**: $80

### 1. **Percentage of buyers who paid less than $1440**:

$1440 is $ \mu - 2\sigma $:
- $1440 = 1600 - 2 \times 80$
- According to the empirical rule, approximately 95% of the data falls within $\mu \pm 2\sigma$. Therefore, about 5% of the data falls outside this range, with half (2.5%) below $1440.
- **Answer**: Approximately **2.5%** of buyers paid less than $1440.

### 2. **Percentage of buyers who paid between $1520 and $1680**:

Both $1520 and $1680 are within $ \mu \pm 1\sigma $:
- $1520 = 1600 - 80$
- $1680 = 1600 + 80$
- According to the empirical rule, approximately 68% of the data falls within $\mu \pm \sigma$.
- **Answer**: Approximately **68%** of buyers paid between $1520 and $1680.

### 3. **Percentage of buyers who paid less than $1360**:

$1360 is $ \mu - 3\sigma $:
- $1360 = 1600 - 3 \times 80$
- According to the empirical rule, approximately 99.7% of the data falls within $\mu \pm 3\sigma$. Therefore, about 0.3% of the data falls outside this range, with half (0.15%) below $1360.
- **Answer**: Approximately **0.15%** of buyers paid less than $1360.

### 4. **Percentage of buyers who paid between $1600 and $1760**:

$1760 is $ \mu + 2\sigma $:
- $1760 = 1600 + 160$
- From $1600 to $1760 covers half of the range between $ \mu $ and $ \mu + 2\sigma $. Since 95% of the data falls within $\mu \pm 2\sigma$, half of this is from $1600 to $1760.
- **Answer**: Approximately **47.5%** of buyers paid between $1600 and $1760.

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Would you like further details on how these percentages were derived, or do you have any other questions?

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Here are five related questions to explore further:
1. What is the Z-score for a price of $1520?
2. What percentage of buyers paid more than $1760?
3. How would the percentage change if the standard deviation increased to $100?
4. What is the probability of a buyer paying exactly $1600?
5. How does the cumulative distribution function relate to these calculations?

**Tip:** When dealing with normal distributions, understanding the Z-score helps in calculating the exact probability of values.

Explore how to calculate approximate percentages of buyers based on the normal distribution and the empirical rule. Answer questions such as the percentage of buyers who paid less than $1440, between $1520 and $1680, less than $1360, and between $1600 and $1760 using statistical principles. Learn the method involving mean and standard deviation to estimate these percentages.

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