Bot GPT 4-o mini To find the percentiles and the median of the normally distributed electricity bills, we will use the properties of the normal distribution.
Given:
- Mean (\( \mu \)) = $110.56
- Standard deviation (\( \sigma \)) = $22.00
(a) Find the 9th percentile of the bill amounts.
To find the 9th percentile, we first find the z-score that corresponds to the 9th percentile. We can use a z-table, or a standard normal distribution calculator.
The z-score corresponding to the 9th percentile is approximately -1.34 (you can find this value using statistical software or z-score tables).
Now we will use the z-score formula to find the corresponding value of the bill:
\[ z = \frac{X - \mu}{\sigma} \]
Substituting the known values:
\[ -1.34 = \frac{X - 110.56}{22} \]
Now, solve for \( X \):
\[ X - 110.56 = -1.34 \times 22 \] \[ X - 110.56 = -29.48 \] \[ X = 110.56 - 29.48 \] \[ X \approx 81.08 \]
Thus, the 9th percentile of the bill amounts is approximately $81.08.
(b) Find the 66th percentile of the bill amounts.
Now let's find the z-score that corresponds to the 66th percentile. The z-score for the 66th percentile is approximately 0.41.
Using the z-score formula:
\[ 0.41 = \frac{X - 110.56}{22} \]
Solving for \( X \):
\[ X - 110.56 = 0.41 \times 22 \] \[ X - 110.56 = 9.02 \] \[ X = 110.56 + 9.02 \] \[ X \approx 119.58 \]
Thus, the 66th percentile of the bill amounts is approximately $119.58.
(c) Find the median of the bill amounts.
In a normal distribution, the median is equal to the mean. Thus, the median of the bill amounts is:
\[ \text{Median} = \mu = 110.56 \]
So the median of the bill amounts is $110.56.
Summary of Results:
- (a) 9th percentile: $81.08
- (b) 66th percentile: $119.58
- (c) Median: $110.56
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