The prices paid for cars can be standardized to a Normal model, with a mean of  $16,000 and a standard deviation of $400.

A group of 6,000 buyers had participated in a study based on purchase price of their car. Using the Empirical Rule determine about how many of them paid between $14,800 and $15,200?

The number of buyers that paid between $14,800 and $15,200 is:

*(Provide your answer as a whole number)*

To solve this problem using the Empirical Rule (also known as the 68-95-99.7 rule), we can follow these steps:

### Step 1: Identify the mean and standard deviation
- Mean ($\mu$) = $16,000
- Standard deviation ($\sigma$) = $400

### Step 2: Calculate the z-scores
The z-score formula is:
\[
z = \frac{X - \mu}{\sigma}
\]
For $X = 14,800$:
\[
z_{1} = \frac{14,800 - 16,000}{400} = \frac{-1,200}{400} = -3
\]
For $X = 15,200$:
\[
z_{2} = \frac{15,200 - 16,000}{400} = \frac{-800}{400} = -2
\]

### Step 3: Use the Empirical Rule
The Empirical Rule tells us that:
- About 68% of data falls within 1 standard deviation from the mean.
- About 95% of data falls within 2 standard deviations from the mean.
- About 99.7% of data falls within 3 standard deviations from the mean.

The z-scores of -3 to -2 represent the range between the second and third standard deviations below the mean.

### Step 4: Estimate the percentage of buyers
The area between 2 and 3 standard deviations from the mean (on one side) represents approximately 2.35% of the data.

### Step 5: Calculate the number of buyers
To find the number of buyers within this range, multiply the total number of buyers by the percentage:
\[
\text{Number of buyers} = 6000 \times 0.0235 = 141 \text{ buyers}
\]

So, approximately **141 buyers** paid between $14,800 and $15,200.

Would you like more details or have any questions?

### Related Questions
1. What is the probability that a randomly selected buyer paid more than $16,400?
2. How many buyers paid between $15,200 and $16,400?
3. What is the z-score for a car priced at $17,000?
4. How many buyers paid less than $14,000?
5. What percentage of buyers paid between $14,000 and $16,800?
6. What would be the percentage of buyers paying above $17,200?
7. Can you determine the number of buyers who paid between $16,000 and $17,600?
8. What is the percentage of buyers paying within one standard deviation of the mean?

### Tip
Remember that the Empirical Rule only applies to data that is normally distributed. Ensure the data follows a normal distribution before applying this rule.

Learn how to estimate the number of car buyers who paid between $14,800 and $15,200 using the Empirical Rule. Detailed step-by-step solution provided with calculations based on a Normal distribution model.

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