Question

The time $X$ it takes Hattan to drive to work on a randomly selected day follows a distribution that is approximately Normal with mean $15$ minutes and standard deviation $6.5$ minutes. Once he parks his car in his reserved space, it takes $5$ more minutes for him to walk to his office. Let $T=$ the total time it takes Hattan to reach his office on a randomly selected day, so $T=X+5$. Describe the shape, center, and variability of the probability distribution of $T$.

Short answer

$T$ is a normal distribution with a mean of $\boxed{\mathbf{20}}$ and a standard deviation of $\boxed{\mathbf{6}\mathbf{.}\mathbf{5}}$ minutes.

Step-by-step solution

Given information

Given:

The time taken by Hattan to drive to work is :$X$

Time taken is Normal with mean : $15$minutes

Standard deviation : $6.5$minutes

It takes for him to walk to his office : $5$minutes

$T=$the total time it takes Hattan to reach his office

$\Rightarrow T=X+5$

Describing the shape, center, and variability of the probability distribution of T

Shape :

$T$is a normal distribution function since adding the constant to every data value has no effect on the shape of the distribution.

Center:

When the constant is added to each data value, the center of the distribution is also enlarged, making $T$a normal distribution function.

$$\begin{gathered} {\mu }_T={\mu }_X+5 \\ =15+5 \\ =\mathbf{20} \end{gathered}$$

Spread:

When you add the constant to every data value, the spread of the distribution does not change; it stays the same.

${\sigma }_T={\sigma }_X=\mathbf{6}\mathbf{.}\mathbf{5}$