A set of final examinations grades in an introductory statis

A set of final examinations grades in an introductory statistics course is normally distributed, with a mean of 75 and a standard deviation of 7. Complete parts (a) through (d).

a.) What is the probability that a student scored below 87 on this exam?

b.) What is the probability that a student scored between 68 and 90?

c.) The probability is 25% that a student taking the test scores higher than what grade?

d.) If the professor grades on a curve (For example, the professor could give A

Solution

mean = 75 and a standard deviation = 7

a) probability that a student scored below 87 = P[X < 87] = P[Z < (87 - 75)/7] = P[Z < 1.714] = 0.9568

b) probability that a student scored between 68 and 90

= P[68 < X < 90]

= P[Z < (90 - 75)/7] - P[Z < (68 - 75)/7]

= P[Z < 2.14] - P[Z < -1]

= 0.9839 - 0.1587

= 0.8253

c) probability 25%

z score for p = 0.25 is -0.6745

-0.6745 = (X - 75)/7

X = 70.28

d) This exam

P[Z = (89 - 75)/7] = P[Z = 2] = 0.9772

mean = 69 and a standard deviation = 3 grade = 72

P[Z = (72 - 69)/3] = P[Z = 1] = 0.8414

A student is worse off with a grade of 89 on this exam because the Z-value for the grade of 89 is   2 and the Z value for the grade of 72 is 1