Question

the amount of time shoppers wait in line can be described by a continuous random variable, x, that is uniformly distributed from 2 to 11 minutes. what is the probability that a randomly chosen shopper will wait in line between 5 and 8 minutes? round to 2 decimal places.

Explanation:

Step1: Determine the formula for uniform distribution probability

For a uniform distribution \(X\sim U(a,b)\), the probability density function is \(f(x)=\frac{1}{b - a}\), and the probability \(P(c<X<d)=\frac{d - c}{b - a}\), where \(a = 2\), \(b=11\), \(c = 5\), \(d = 8\).

Step2: Substitute values into the formula

Substitute \(a = 2\), \(b = 11\), \(c = 5\), \(d = 8\) into \(P(c<X<d)=\frac{d - c}{b - a}\).
We get \(P(5<X<8)=\frac{8 - 5}{11 - 2}\).

Step3: Calculate the result

\(\frac{8 - 5}{11 - 2}=\frac{3}{9}=\frac{1}{3}\approx0.33\)

Answer:

\(0.33\)