The problem relates to a normal distribution for the prices paid for an HD television, with the following given values:

• Mean price ($\mu$) = $1000
• Standard deviation ($\sigma$) = $100

The questions involve calculating approximate percentages for various ranges using the properties of the standard normal distribution (Z-scores).

Let’s break down each part:

1. What is the approximate percentage of buyers who paid more than $1300?

$Z = \frac{1300 - 1000}{100} = 3$ From standard normal distribution tables, the probability corresponding to $Z = 3$ is approximately $0.9987$. The percentage beyond $Z = 3$ is: $1 - 0.9987 = 0.0013 = 0.13\%$ Answer: 0.13%

2. What is the approximate percentage of buyers who paid less than $800?

$Z = \frac{800 - 1000}{100} = -2$ From standard normal distribution tables, the probability corresponding to $Z = -2$ is approximately $0.0228$. This gives: $2.28\%$ Answer: 2.28%

3. What is the approximate percentage of buyers who paid between $900 and $1100?

For $900$: $Z = \frac{900 - 1000}{100} = -1$ The probability for $Z = -1$ is approximately $0.1587$.

For $1100$: $Z = \frac{1100 - 1000}{100} = 1$ The probability for $Z = 1$ is approximately $0.8413$.

The percentage between $900 and $1100 is: $0.8413 - 0.1587 = 0.6826 = 68.26\%$ Answer: 68.26%

4. What is the approximate percentage of buyers who paid between $900 and $1000?

For $900$ ($Z = -1$): $P(Z < -1) = 0.1587$ For $1000$ ($Z = 0$): $P(Z < 0) = 0.5$

The percentage is: $0.5 - 0.1587 = 0.3413 = 34.13\%$ Answer: 34.13%

5. What is the approximate percentage of buyers who paid between $1000 and $1300?

For $1000$ ($Z = 0$): $P(Z < 0) = 0.5$ For $1300$ ($Z = 3$): $P(Z < 3) = 0.9987$

The percentage is: $0.9987 - 0.5 = 0.4987 = 49.87\%$ Answer: 49.87%

6. What is the approximate percentage of buyers who paid between $1000 and $1200?

For $1000$ ($Z = 0$): $P(Z < 0) = 0.5$ For $1200$ ($Z = 2$): $P(Z < 2) = 0.9772$

The percentage is: $0.9772 - 0.5 = 0.4772 = 47.72\%$ Answer: 47.72%

Let me know if you need further clarification or explanation!

Here are 5 relative questions to expand your understanding:

1. What are the key properties of the standard normal distribution?
2. How do you calculate a Z-score, and what does it represent?
3. Why is the area under the curve in a normal distribution equal to 1?
4. How would the results change if the standard deviation were larger (e.g., $200)?
5. What is the relationship between empirical rules (68-95-99.7) and Z-scores?

Tip: When solving normal distribution problems, always sketch the curve and mark the regions you're calculating—this visual aid helps avoid errors!