the average yearly utility costs on Middleton is $1722 with stdev of 146. assume that the utility costs follow a normal distribution.
1) What would the utility costs have to be so that only 7% of houses in Middleton have utility costs greater than this amount
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.07) and its Z score. Insert Z score in equation above to find raw score.
To find the utility cost threshold such that only 7% of houses in Middleton have costs greater than that amount, we need to use the z-score formula and the standard normal distribution table.
1. First, we need to find the z-score corresponding to the desired percentile (7%). We can find this using a standard normal distribution table or a calculator.
The formula for calculating the z-score is:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the utility cost we want to find
- μ is the mean of the utility costs (1722)
- σ is the standard deviation of the utility costs (146)
2. In this case, we want to find the utility cost (x) such that only 7% of houses have costs greater than that amount. This means we need to find the z-score that corresponds to a cumulative probability of 1 - 0.07 = 0.93 (7% of houses have costs lower than the threshold).
3. Using a standard normal distribution table (or a calculator), we can find that the z-score corresponding to a cumulative probability of 0.93 is approximately 1.48.
4. Now, we can plug the values into the formula and solve for x:
1.48 = (x - 1722) / 146
5. Rearranging the equation to solve for x:
x - 1722 = 1.48 * 146
x - 1722 = 215.68
x = 1722 + 215.68
x ≈ 1937.68
So, the utility costs would have to be approximately $1937.68 for only 7% of houses in Middleton to have utility costs greater than that amount.