1 Answer

Arganzheng · Answer 1 · 2021-11-26T05:14:25+0000

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.

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:

$f_(X)(x)=(1)/(b-a);\ a<X<b,\ a<b$

(a)

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

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

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:

$P(35<X<42)=\int\limits^(42)_(35) {(1)/(55-31)}}\, dx=(1)/(24) |x|^(42)_(35)=(42-35)/(24)=0.2917$

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)

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

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

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