Question

Total gross profits $G$ on a randomly selected day at Tim’s Toys follow a distribution that is approximately Normal with mean $\backslash (560$ and standard deviation $\backslash )185$. The cost of renting and maintaining the shop is $\$65$ per day. Let $P=$profit on a randomly selected day, so $P=G-65$. Describe the shape, center, and variability of the probability distribution of $P$.

Short answer

$P$ is a normal distribution with a mean of $\$495$ and a standard deviation of $\$185.$

Step-by-step solution

Given information

Given :

Total gross profits on a randomly selected day : $G$

It is approximately Normal with mean :$\$560$

Standard deviation : $\$185$.

The cost of renting and maintaining the shop is :$\$65$

Let$P=$profit on a randomly selected day

$\Rightarrow P=G-65.$

Describing the shape, center, and variability of the probability distribution of P.

Shape :

$P$is a normal distribution function because subtracting the constant from each data value has no effect on the shape of the distribution.

Center :

$P$is a normal distribution function because when the constant is subtracted from each data value, the center of the distribution is also decreased by that constant value.

$$\begin{gathered} {\mu }_p={\mu }_G–65 \\ =560–65 \\ =\mathbf{495} \end{gathered}$$

Spread :

Subtracting the constant from each data value has no effect on the distribution's spread; it remains unaffected.

${\sigma }_p={\sigma }_g=\mathbf{185}$