Final answer:
The probability density function for the length of time it takes students to complete a statistics examination that is uniformly distributed between 50 and 75 minutes is 1/25. The probability that a student will take between 55 and 65 minutes to complete the examination is 0.4. The probability that a student will take no more than 50 minutes to complete the examination is 0.
Step-by-step explanation:
(a) The probability density function (PDF) is given by:
f(x) = 1/ (b - a) = 1/ (75 - 50) = 1/ 25
(b) To find the probability that a student will take between 55 and 65 minutes, we need to calculate the area under the probability density function curve between these two points. The probability is given by:
P(55 ≤ x ≤ 65) = (65 - 55) / (75 - 50) = 10 / 25 = 0.4
(c) To find the probability that a student will take no more than 50 minutes, we need to calculate the area under the probability density function curve up to 50 minutes. The probability is given by:
P(x ≤ 50) = (50 - 50) / (75 - 50) = 0 / 25 = 0
(d) The expected amount of time it takes a student to complete the examination is given by the mean of the probability density function, which is the average of the minimum and maximum values:
E(X) = (a + b) / 2 = (50 + 75) / 2 = 62.5
(e) The variance in the amount of time it takes a student to complete the examination is given by the formula:
Var(X) = [(b - a)^2] / 12 = [(75 - 50)^2] / 12 = 625 / 12 ≈ 52.08