To find the amount a family would have spent to be in the 80th percentile, we can use the formula for a value in a normal distribution:

$X = \mu + Z \cdot \sigma$

Where:

• $X$ is the value corresponding to the 80th percentile.
• $\mu$ is the mean of the distribution (given as $1,600).
• $Z$ is the Z-score for the 80th percentile (we can look this up from a Z-table or calculate using a statistical tool).
• $\sigma$ is the standard deviation (given as $411).

From a Z-table or calculator, the Z-score corresponding to the 80th percentile is approximately 0.8416.

Now, applying the formula:

$X = 1600 + (0.8416 \times 411)$

Let's compute this value.The amount a family would have spent to be in the 80th percentile is approximately $1,945.90.

Do you want more details or have any questions? Here are 5 related questions to expand on this:

1. How would the amount change if the standard deviation increased?
2. How is the Z-score calculated for different percentiles?
3. What is the 90th percentile for this distribution?
4. How does the normal distribution assumption affect this calculation?
5. Can this method be applied to other types of spending, like annual grocery bills?

Tip: In a normal distribution, the higher the standard deviation, the wider the spread of data points, meaning percentiles will cover a larger range.