2. The exam scores of a course follow a normal distribution with a mean of 78 an

Question

2. The exam scores of a course follow a normal distribution with a mean of 78 and a standard deviation of 11.
(a) We are interested in the probability of a student getting a C or better (score of 70 or more). Based on the description of the problem, should this probability be greater than 0.5 or lower than 0.5? (COMPUTATION NOT NECESSARY HERE)
(b) Find the probability that a student gets a C or better in the exam.
(c) What would be the grade of a student such that she is the 20th percentile?
(d) What would be the grade of a student such that only 5% of the students obtain a better grade. 2. The exam scores of a course follow a normal distribution with a mean of 78 and a standard deviation of 11.
(a) We are interested in the probability of a student getting a C or better (score of 70 or more). Based on the description of the problem, should this probability be greater than 0.5 or lower than 0.5? (COMPUTATION NOT NECESSARY HERE)
(b) Find the probability that a student gets a C or better in the exam.
(c) What would be the grade of a student such that she is the 20th percentile?
(d) What would be the grade of a student such that only 5% of the students obtain a better grade. 2. The exam scores of a course follow a normal distribution with a mean of 78 and a standard deviation of 11.
(a) We are interested in the probability of a student getting a C or better (score of 70 or more). Based on the description of the problem, should this probability be greater than 0.5 or lower than 0.5? (COMPUTATION NOT NECESSARY HERE)
(b) Find the probability that a student gets a C or better in the exam.
(c) What would be the grade of a student such that she is the 20th percentile?
(d) What would be the grade of a student such that only 5% of the students obtain a better grade.

Explanation / Answer

a) The probability will be greater than 0.5.

b) P(X > 70) = P((X - mean)/sd > (70 - mean)/sd)

                    = P(Z > (70 - 78)/11)

                    = P(Z > -0.73)

                    = 1 - P(Z < -0.73)

                    = 1 - 0.2327 = 0.7673

b) P(X < x) = 0.2

or, P((X - mean)/sd < (x - mean)/sd) = 0.2

or, P(Z < (x - 78)/11) = 0.2

or, (x - 78)/11 = -0.84

or, x = -0.84 * 11 + 78

or, x = 68.76

d) P((X > x) = 0.05

or, P((X - mean)/sd > (x - 78)/11) = 0.05

or, P(Z > (x - 78)/11) = 0.05

or, P(Z < (x - 78)/11) = 0.95

or, (x - 78)/11 = 1.645

or, x = 1.645 * 11 + 78

or, x = 96.095

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