Question

Suppose commute times in a large city are normally distributed and that 65.20% of commuters in this city take more than 20 minutes to commute one-way. If the standard deviation of such commutes is 6.3 minutes, what is the mean commute?

Answer 1

Answer:

The mean time for commute is 22.46 minutes.

Step-by-step explanation:

We are given the following information in the question:

Standard Deviation, σ = 6.3 minutes

We are given that the distribution of commute times in a large city is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

P(more than 20 minutes to commute one-way)  = 65.20% = 0.6520

P(X > 20) = 0.6520

$P( x > 20) = P( z > \displaystyle\frac{20 - \mu}{6.3}) = 0.6520$

$= 1 - P(z \leq \displaystyle\frac{20 - \mu}{6.3}) = 0.6520$

$P(z \leq \displaystyle\frac{20 - \mu}{6.3}) =1-0.6520=0.348$

Calculation the value from standard normal z table, we have,  

$P(z \leq -0.391) =0.348$

$\displaystyle\frac{20 - \mu}{6.3} = -0.391$

Solving, we get,

$\mu = 22.46\text{ minutes}$

The mean time for commute is 22.46 minutes.