Let's solve this step-by-step while elaborating on each part of the question in detail.
Given:
- The mean monthly household electricity bill in the United States in 2011: \[tex]$109.72
- Standard deviation: \$[/tex]17.00
We need to find the following:
(a) The 7th percentile of the bill amounts.
(b) The 64th percentile of the bill amounts.
(c) The median of the bill amounts.
### Part 1: The 7th percentile of the bill amounts
To find the 7th percentile, we need to determine the point at which 7% of the values lie below it in a normal distribution with the given mean and standard deviation.
The value corresponding to the 7th percentile is found to be approximately:
[tex]\[ \$84.63 \][/tex]
### Part 2: The 64th percentile of the bill amounts
To find the 64th percentile, we need to determine the point at which 64% of the values lie below it in the normal distribution with the given mean and standard deviation.
The value corresponding to the 64th percentile is:
[tex]\[ \$115.81 \][/tex]
### Part 3: The median of the bill amounts
In a normal distribution, the median is the same as the mean. Therefore, the median of the bill amounts is:
[tex]\[ \$109.72 \][/tex]
### Summary of Answers:
(a) The 7th percentile of the bill amounts is: \[tex]$84.63
(b) The 64th percentile of the bill amounts is: \$[/tex]115.81
(c) The median of the bill amounts is: \$109.72
By carefully assessing each part and utilizing the properties of the normal distribution, we have determined the required percentiles and the median.
