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 <em>X</em> = amount of time it takes for a student to complete a statistics quiz.
The random variable <em>X</em> is uniformly distributed with parameters <em>a</em> = 31 minutes and <em>b</em> = 55 minutes.
The probability density function of <em>X</em> is:
![f_{X}(x)=\frac{1}{b-a};\ a](https://tex.z-dn.net/?f=f_%7BX%7D%28x%29%3D%5Cfrac%7B1%7D%7Bb-a%7D%3B%5C%20a%3CX%3Cb%2C%5C%20a%3Cb)
(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](https://tex.z-dn.net/?f=P%28X%3E51%29%3D%5Cint%5Climits%5E%7B55%7D_%7B51%7D%20%7B%5Cfrac%7B1%7D%7B55-31%7D%7D%7D%5C%2C%20dx%3D%5Cfrac%7B1%7D%7B24%7D%20%7Cx%7C%5E%7B55%7D_%7B51%7D%3D%5Cfrac%7B55-51%7D%7B24%7D%3D0.1667)
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](https://tex.z-dn.net/?f=P%2835%3CX%3C42%29%3D%5Cint%5Climits%5E%7B42%7D_%7B35%7D%20%7B%5Cfrac%7B1%7D%7B55-31%7D%7D%7D%5C%2C%20dx%3D%5Cfrac%7B1%7D%7B24%7D%20%7Cx%7C%5E%7B42%7D_%7B35%7D%3D%5Cfrac%7B42-35%7D%7B24%7D%3D0.2917)
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](https://tex.z-dn.net/?f=%3D%5Cint%5Climits%5E%7B42.34%7D_%7B41.34%7D%20%7B%5Cfrac%7B1%7D%7B55-31%7D%7D%7D%5C%2C%20dx%3D%5Cfrac%7B1%7D%7B24%7D%20%7Cx%7C%5E%7B42.34%7D_%7B41.34%7D%3D%5Cfrac%7B42.34-41.34%7D%7B24%7D%3D0.0417)
Thus, the probability that a student completes the quiz in exactly 41.84 minutes is 0.0417.