The graph illustrates a normal distribution for the prices paid for a particular model of HD television. The mean price paid is $1600 and the standard deviation is $80.

1360144015201600168017601840Distribution of Prices


To analyze the normal distribution of the HD television prices based on the provided information, we can break down the key components:

1. **Mean ($\mu$)**: The average price paid is $1600.
2. **Standard Deviation ($\sigma$)**: The standard deviation is $80.

### Interpreting the Normal Distribution:
A normal distribution is symmetric around its mean. The prices are spread out with a bell-shaped curve, and the standard deviation helps us understand the spread of the data around the mean.

#### Important Intervals:
- **Within 1 standard deviation ($\mu \pm \sigma$)**:
  - Range: $1600 $\pm$ 80 = [\$1520, \$1680]$
  - Approximately 68% of the prices paid fall within this range.
  
- **Within 2 standard deviations ($\mu \pm 2\sigma$)**:
  - Range: $1600 $\pm$ 160 = [\$1440, \$1760]$
  - Approximately 95% of the prices fall within this range.
  
- **Within 3 standard deviations ($\mu \pm 3\sigma$)**:
  - Range: $1600 $\pm$ 240 = [\$1360, \$1840]$
  - Approximately 99.7% of the prices fall within this range.

### Visual Representation:
- The curve is centered at $1600.
- It extends from approximately $1360 to $1840, covering almost all possible prices.

### Application:
Knowing these intervals allows you to estimate the likelihood of a given price. For example, if someone paid $1680, they are within 1 standard deviation of the mean, meaning they paid a price that is quite common.

### Next Steps:
Would you like to dive deeper into any particular aspect of this distribution, such as calculating probabilities or exploring how prices compare to the mean?

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Here are five related questions to explore further:
1. What is the probability of paying more than $1680 for the HD television?
2. How do you calculate the Z-score for a price of $1440?
3. What percentage of prices fall between $1520 and $1760?
4. If the standard deviation increased, how would that affect the spread of prices?
5. What is the significance of the area under the curve in a normal distribution?

**Tip:** In a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution.

Explore the normal distribution of HD television prices with a mean of $1600 and a standard deviation of $80. Understand key intervals within the distribution and their implications. Suitable for advanced learners and professionals.

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