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Answer:
(a) The probability distribution of the random variable <em>X</em> is Binomial.
(b) The mean and standard deviation of the 25-question test are 5 and 2 respectively.
(c) The value of <em>x</em> is 0.34.
Step-by-step explanation:
The random variable <em>X </em>is defined as the number of correct answers given by a person who is guessing each answer on a 25-question exam.
There are five possible answer for every question.
This implies that the probability of getting a correct answer is:
<em>P</em> (X) = 0.20.
There are a total of <em>n</em> = 25 questions.
Every answer is independent of the others.
(a)
The random variable <em>X</em> has finite number of independent trials (i.e. 25 questions). There are only two outcomes for each trial, i.e. Success = correct answer and Failure = wrong answer. Each trial has the same probability of success (, i.e. P (X) = 0.25).
Thus, the probability distribution of the random variable <em>X</em> is Binomial with parameters <em>n</em> = 25 and <em>p</em> = 0.20.
(b)
Compute the mean of the random variable <em>X</em> as follows:
Compute the standard deviation of the random variable <em>X</em> as follows:
Thus, the mean and standard deviation of the 25-question test are 5 and 2 respectively.
(c)
The sample is large and the probability of success is close to 0.50.
So a Normal approximation to binomial can be applied to approximate the distribution of X if the following conditions are satisfied:
1. np ≥ 5
2. n(1 - p) ≥ 5
Check the conditions as follows:
Thus, a Normal approximation to binomial can be applied.
So,
It is provided that the minimum passing score foe the test is such that only 1% of students who are guessing will pass the test.
That is P (X < x) = 0.01.
⇒ P (Z < z) = 0.01
The value of <em>z</em> is -2.33.
Compute the value of <em>x</em> as follows:
Thus, the value of <em>x</em> is 0.34.
Answer: The length of the parking lot should be increased 20 ft.
Please, see the attached files.
Thanks.
