Question

Los Angeles workers have an average commute of 29 minutes. Suppose the LA commute time is normally distributed with a standard deviation of 13 minutes. Let X represent the commute time for a randomly selected LA worker. Round all answers to two decimal places.

A. X ~ N(______ , ______ )

B. Find the probability that a randomly selected LA worker has a commute that is longer than 40 minutes. __________

C.Find the 90th percentile for the commute time of LA workers. ________

Answer 1

The commute time for LA workers is represented by X ~ N(29, 13). The probability of a commute longer than 40 minutes is approximately 0.1977. The 90th percentile can be found using a standard normal distribution table and converting the Z-score back to the actual commute time.

Step-by-step explanation:

Calculating Probabilities and Percentiles for LA Workers' Commute Time

A. Given that Los Angeles workers have an average commute of 29 minutes and a standard deviation of 13 minutes, we can represent the commute time X for a randomly selected LA worker with a normal distribution: X ~ N(29, 13).

B. To find the probability that a randomly selected LA worker has a commute that is longer than 40 minutes, we use the standard normal distribution (Z-score) formula: Z = (X - μ) / σ, where X is the commute time, μ is the mean, and σ is the standard deviation. Plugging in the values gives Z = (40 - 29) / 13 ≈ 0.85. Using a Z-table or calculator, we find the probability associated with Z > 0.85. The area to the right of Z is 1 - the cumulative probability of Z, therefore, P(X > 40) ≈ 1 - 0.8023 = 0.1977.

Answer 2

a)
`X\sim (29,169)`

b) 0.1987

c)
`P_(70)=35.82`

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 29 minutes

Standard Deviation, σ = 13 minutes

We are given that the distribution of commute time is a bell shaped distribution that is a normal distribution.

a) Distribution of X

Let X represent the commute time for a randomly selected LA worker. Then,

`X\sim (\mu, \sigma^2)\\X\sim(29,(13)^2)\\X\sim (29,169)`

b) Probability that a randomly selected LA worker has a commute that is longer than 40 minutes

`P( x > 40) = P( z > \displaystyle(40 - 29)/(13)) = P(z > 0.8461)`

`= 1 - P(z \leq 0.8461)`

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

`P(x > 40) = 1 - 0.8013 =0.1987`

c) 70th percentile for the commute time of LA workers.

We have to find the value of x such that the probability is 0.7

`P( X < x) = P( z < \displaystyle(x - 29)/(13))=0.7`

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

`\displaystyle(x - 29)/(13) = 0.524\\\\x = 35.812\approx 35.82`

The 70th percentile for the distribution of commute time of LA workers is 35.81 minutes.