Question

Suppose the average speeds of passenger trains traveling from Newark, New Jersey, to Philadelphia, Pennsylvania, are normally distributed, with a mean average speed of 87 miles per hour and a standard deviation of 6.4 miles per hour. (a) What is the probability that a train will average less than 73 miles per hour? (b) What is the probability that a train will average more than 80 miles per hour? (c) What is the probability that a train will average between 90 and 99 miles per hour?

(a) P(x < 73)

(b) P(x > 80)

(c) P(90 ≤ x ≤ 99)

Answer 1

Solution :

Let X be a random variable which represents the average speed of trains running from Newark, New Jersey, to Philadelphia, Pennsylvania.

Given that, X ~ N(87, 6.42)

i.e.  μ = 87 miles/hour and σ = 6.4 miles/hour

a) We have to obtain P(X < 73).

We know that if X ~ N(μ, σ​​​​​​2) then,

Using "pnorm" function of R we get, P(Z < -2.1875) = 0.0144

The probability that a train will average less than 73 miles per hour is 0.0144.

b) We have to obtain P(X > 80).

We know that if X ~ N(μ, σ​​​​​​2) then,

Using "pnorm" function of R we get, P(Z > -1.09375) = 0.8630

The probability that a train will average more than 80 miles per hour is 0.8630.

c) We have to obtain P(90 ≤ x ≤ 99).

We know that if X ~ N(μ, σ​​​​​​2) then,

Using "pnorm" function of R we get,

P(Z < 1.875) = 0.9696 and P(Z < 0.46875) = 0.6804

The probability that a train will average between 90 and 99 miles per hour is 0.2892.

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