Question

a statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed between 50.0 and 60.0 minutes. find the probability that a given class period runs between 51.5 and 51.75 minutes.
find the probability of selecting a class that runs between 51.5 and 51.75 minutes.
(round to three decimal places as needed.)

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 \(a = 50.0\), \(b=60.0\), \(c = 51.5\), \(d = 51.75\).

Step2: Substitute the values into the formula

Substitute \(a = 50.0\), \(b = 60.0\), \(c = 51.5\), \(d = 51.75\) into \(P(c<X<d)=\frac{d - c}{b - a}\).
We get \(P(51.5<X<51.75)=\frac{51.75 - 51.5}{60.0 - 50.0}\).
First, calculate the numerator: \(51.75-51.5 = 0.25\).
Then, calculate the denominator: \(60.0 - 50.0=10.0\).
So, \(P(51.5<X<51.75)=\frac{0.25}{10.0}=0.025\).

Answer:

\(0.025\)