Question

The amount of time it takes for a student to complete a statistics quiz is uniformly distributed (or, given by a random variable that is uniformly distributed) between 31 and 55 minutes. One student is selected at random. Find the probability of the following events.A. The student requires more than 51 minutes to complete the quiz.Probability =B. The student completes the quiz in a time between 35 and 42 minutes.Probability =C. The student completes the quiz in exactly 41.84 minutes.Probability =

Answer 1

Answer:

(a) The probability that a student requires more than 51 minutes to complete the quiz is 0.1667.

(b) The probability that a student completes the quiz in a time between 35 and 42 minutes is 0.2917.

(c) The probability that a student completes the quiz in exactly 41.84 minutes is 0.0417.

Step-by-step explanation:

Let X = amount of time it takes for a student to complete a statistics quiz.

The random variable X is uniformly distributed with parameters a = 31 minutes and b = 55 minutes.  

The probability density function of X is:

$f_{X}(x)=\frac{1}{b-a};\ a$

(a)

Compute the probability that a student requires more than 51 minutes to complete the quiz as follows:

$P(X>51)=\int\limits^{55}_{51} {\frac{1}{55-31}}}\, dx=\frac{1}{24} |x|^{55}_{51}=\frac{55-51}{24}=0.1667$

Thus, the probability that a student requires more than 51 minutes to complete the quiz is 0.1667.

(b)

Compute the probability that a student completes the quiz in a time between 35 and 42 minutes as follows:

$P(35$

Thus, the probability that a student completes the quiz in a time between 35 and 42 minutes is 0.2917.

(c)

Compute the probability that a student completes the quiz in exactly 41.84 minutes as follows:

Apply continuity correction as follows:

P (X = 41.84) = P (41.84 - 0.50 < X < 41.84 + 0.50)

                     = P (41.34 < X < 42.34)

$=\int\limits^{42.34}_{41.34} {\frac{1}{55-31}}}\, dx=\frac{1}{24} |x|^{42.34}_{41.34}=\frac{42.34-41.34}{24}=0.0417$

Thus, the probability that a student completes the quiz in exactly 41.84 minutes is 0.0417.