Question

thanks to social media, the happiest creature on earth, an always - smiling australian marsupial called a quokka, has become well - known. suppose that weights of quokkas can be described by a normal model with a mean of 9 pounds and a standard deviation of 2.3 pounds.
a) a zoo keeper keeps a record of the weights of quokka in her zoo. if she subtracts the mean of 9 pounds from all the weights, what is the mean and standard deviation of the remainders?
b) suppose transporting such small animals costs $1.50 per pound. what is the mean and standard deviation of the cost of transport?
a) the new mean would be pound(s). the new standard deviation would be pound(s). (type integers or decimals. do not round.)

Explanation:

Step1: Recall properties of mean and standard - deviation for mean - centering

When we subtract the mean $\mu$ from each data - point $x_i$ in a data set, the new data set $y_i=x_i - \mu$. The formula for the mean of the new data set $\bar{y}$ is $\bar{y}=\frac{\sum_{i = 1}^{n}(x_i-\mu)}{n}=\frac{\sum_{i = 1}^{n}x_i}{n}-\mu$. Since $\frac{\sum_{i = 1}^{n}x_i}{n}=\mu$, the new mean $\bar{y}=0$. The standard deviation measures the spread of the data. Subtracting a constant from each data - point does not change the spread. So, if the original standard deviation is $\sigma$, the new standard deviation is still $\sigma$.
Let the original weights of quokkas have mean $\mu = 9$ pounds and standard deviation $\sigma=2.3$ pounds. After subtracting the mean from all the weights, the new mean is $0$ pounds and the new standard deviation is $2.3$ pounds.

Step2: Recall properties of mean and standard - deviation for linear transformation

If we have a linear transformation $y = ax$ where $a$ is a constant, the mean of $y$ is $\mu_y=a\mu_x$ and the standard deviation of $y$ is $\sigma_y = |a|\sigma_x$. Here, the cost of transporting an animal is $y = 1.5x$ where $x$ is the weight of the animal in pounds. The original mean weight is $\mu_x = 9$ pounds and the original standard deviation is $\sigma_x=2.3$ pounds.
The new mean cost $\mu_y=1.5\times9 = 13.5$ dollars and the new standard deviation of the cost $\sigma_y=1.5\times2.3 = 3.45$ dollars.

Answer:

a) The new mean would be $0$ pound(s). The new standard deviation would be $2.3$ pound(s).
b) The mean of the cost of transport is $13.5$ dollars. The standard deviation of the cost of transport is $3.45$ dollars.