Final answer:
To find the utility bill that cuts off the top 20%, you would use the z-score for the 80th percentile and the formula for a value in a normal distribution. A z-score of approximately 0.84 corresponds to the top 20%, and using the formula X = μ + (z * σ), the utility bill amount is approximately $110.08.
Step-by-step explanation:
The question asks to find the monthly utility bill that cuts off the top 20% of the data, given that the bills are normally distributed with a mean of $100 and a standard deviation of $12. To solve this, one would use the concept of z-scores and the standard normal distribution. The z-score corresponding to the top 20% cut-off point can be found using standard normal distribution tables or a calculator capable of inverse normal probability calculations.
For example, a z-score that cuts off the top 20% often corresponds to a z-score of approximately 0.84 when looking at a standard normal distribution table. To find the particular utility bill value, you can use the formula:
X = μ + (z * σ)
Where:
- X is the bill amount that cuts off the top 20%
- μ (mu) is the mean of the utility bills ($100)
- z is the z-score (0.84 for the top 20%)
- σ (sigma) is the standard deviation of the utility bills ($12)
By plugging in the values, the calculation for X would be:
X = $100 + (0.84 * $12) = $100.08
Therefore, a utility bill of approximately $110.08 cuts off the top 20% of the data.