Questions: Based on accumulated data, a propane gas company determined that the random variable X, the number of gallon required to fill a customer's propane tank, has mean μx=300 gallons and standard deviation σx=40 gallons. The company charges the customer 25 service fee plus 2.50 per gallon of propone. What is the mean and standard deviation of the amount that a customer is billed for? mean = 750, standard deviation = 100 mean = 775, standard deviation = 125 mean = 125, standard deviation = 750 mean = 775, standard deviation = 100 mean = 750. standard deviation = 125

Based on accumulated data, a propane gas company determined that the random variable X, the number of gallon required to fill a customer's propane tank, has mean μx=300 gallons and standard deviation σx=40 gallons. The company charges the customer 25 service fee plus 2.50 per gallon of propone.

What is the mean and standard deviation of the amount that a customer is billed for?
mean = 750, standard deviation = 100
mean = 775, standard deviation = 125
mean = 125, standard deviation = 750
mean = 775, standard deviation = 100
mean = 750. standard deviation = 125

Transcript text: Based on accumulated data, a propane gas company determined that the random variable $X$, the number of gallon required to fill a customer's propane tank, has mean $\mu_{\mathrm{x}}=300$ gallons and standard deviation $\sigma_{\mathrm{x}}=40$ gallons. The company charges the customer $\$ 25$ service fee plus $\$ 2.50$ per gallon of propone. What is the mean and standard deviation of the amount that a customer is billed for? mean $=\$ 750$, standard deviation $=\$ 100$ mean $=\$ 775$, standard deviation $=\$ 125$ mean $=\$ 125$, standard deviation $=\$ 750$ mean $=\$ 775$, standard deviation $=\$ 100$ mean $=\$ 750$. standard deviation $=\$ 125$

Solution

Solution Steps

To determine the mean and standard deviation of the amount a customer is billed, we need to consider the billing formula: $B = 25 + 2.50X$. Given the mean and standard deviation of $X$, we can use the properties of linear transformations of random variables to find the mean and standard deviation of $B$.

The mean of $B$ can be found using the formula: $\mu_B = a + b\mu_X$, where $a$ is the fixed service fee and $b$ is the cost per gallon.
The standard deviation of $B$ can be found using the formula: $\sigma_B = |b|\sigma_X$.

Step 1: Calculate the Mean of the Amount Billed

Given the mean of the random variable $X$, $\mu_X = 300$ gallons, the service fee of \$25, and the cost per gallon of \$2.50, we can calculate the mean of the amount billed, $\mu_B$, using the formula: \[ \mu_B = a + b\mu_X \] where $a = 25$ and $b = 2.50$. Substituting the values, we get: \[ \mu_B = 25 + 2.50 \times 300 = 25 + 750 = 775 \]

Step 2: Calculate the Standard Deviation of the Amount Billed

Given the standard deviation of the random variable $X$, $\sigma_X = 40$ gallons, and the cost per gallon of \$2.50, we can calculate the standard deviation of the amount billed, $\sigma_B$, using the formula: \[ \sigma_B = |b|\sigma_X \] where $b = 2.50$. Substituting the values, we get: \[ \sigma_B = 2.50 \times 40 = 100 \]