The results from a statistics class’ first exam are as follows: The average grad

Question

The results from a statistics class’ first exam are as follows: The average grade obtained on the exam by its 25 students is an 83, with a standard deviation of 11 points. Answer the following based on this information.

a. Approximately how many people received a failing grade (less than 65)?

b. What percentage of people received a grade between a 70 and a 91?

c. What percentage of individuals received a score whose z-score was -.70 or less?

d. What grade is required in order to be in the top 15 percent? The top 10 percent?

e. What percentage of people received a grade between 85 and 95?

f. What percentage of people received a grade of 94 or less?

g. What grade is required to be in the bottom 20%?

h. What z-score is required to be in the top 40%?

i. What percentage of individuals have a z-score between -1 and 1.40?

j. What percentage of individuals have a z-score between 1.05 and 1.40?

Explanation / Answer

a) P(X < 65)

= P((X - mean)/(65 - 83)/11)

= P(Z < -1.64)

= 0.0505

no of people = 0.0505 * 25 = 1

b) P(70 < X < 91)

= P((70 - 83)/11 < Z < (91 - 83)/11)

= P(-1.18 < Z < 0.73)

= P(Z < 0.73) - P(Z < -1.18)

= 0.7673 - 0.1190

= 0.6483

C) P(Z < -0.7) = 0.2420

d) P(X > x) = 0.15

or, P(Z > (x 83)/11) = 0.15

or, P(Z < (x - 83)/11) = 0.85

or, (x - 83)/11 = 1.04

or, x = 1.04 * 11 + 83

or, x = 94.44

P(X > x) = 0.1

or, P(Z > (x 83)/11) = 0.1

or, P(Z < (x - 83)/11) = 0.1

or, (x - 83)/11 = 1.28

or, x = 1.28 * 11 + 83

or, x = 97.08

e) P(85 < X < 95)

= P((85 - 83)/11 < Z < (95 - 83)/11)

= P(0.18 < Z < 1.09)

= P(Z < 1.09) - P(Z < 0.18)

= 0.8621 - 0.5714

= 0.2907

f) :P(X < 94)

= P(Z < (94 - 83)/11)

= P(Z < 1)

= 0.8413

g) P(X < x) = 0.2

or, P(Z < (x - 83)/11) = 0.2

or, (x - 83)/11 = -0.84

or, x = -0.84 * 11 + 83

or, x = 73.76

h) P(X > x) = 0.4

or, P(Z > (x - 83)/11) = 0.4

P(Z < (x - 83)/11) = 0.6

or, (x - 83)/11 = 0.25

or, x = 0.25 * 11 + 83

or, x = 85.75

i) P(-1 < Z < 1.4)

= P(Z < 1.4) - P(Z < -1)

= 0.9192 - 0.1587

= 0.7605

j) P(1.05 < Z < 1.4)

= P(Z < 1.4) - P(Z < 1.05)

= 0.9192 - 0.8531

= 0.0661

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