Question

According to a government energy agency, the mean monthly household electricity bill in the United States in 2011 was $110.56. Assume the amounts are normally distributed with standard deviation $22.00.

(a) Find the 9th percentile of the bill amounts.

(b) Find the 66th percentile of the bill amounts.

(c) Find the median of the bill amounts.

Answer 1

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**