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.