Respuesta :
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:
[tex]f_{X}(x)=\frac{1}{b-a};\ a<X<b,\ a<b[/tex]
(a)
Compute the probability that a student requires more than 51 minutes to complete the quiz as follows:
[tex]P(X>51)=\int\limits^{55}_{51} {\frac{1}{55-31}}}\, dx=\frac{1}{24} |x|^{55}_{51}=\frac{55-51}{24}=0.1667[/tex]
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:
[tex]P(35<X<42)=\int\limits^{42}_{35} {\frac{1}{55-31}}}\, dx=\frac{1}{24} |x|^{42}_{35}=\frac{42-35}{24}=0.2917[/tex]
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)
[tex]=\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[/tex]
Thus, the probability that a student completes the quiz in exactly 41.84 minutes is 0.0417.
