Question

3. (4 points) The scores on a test are normally distributed with a mean of 75 and a standard deviation of 8. a) Find the proportion of students having scores greater than 85. b) If the bottom 3% of students will fail the course, what is the lowest score that a student can have and still be awarded a passing grade? Please round up to the nearest integer.

Answer 1

- Let X be the test scores as random variable Then, XN N(75, 82 (@) P(x285) pl X-75 y 85-75 8 P(Z > 1.25) - PLZL 1.25) 1- 0.89435 = 0.10565 (6) Let y be lowest passing score. Then, P(X2) 0.03 y-75 3) 0.03 P(Z & 9-75 0.03 Using 2-tables: P(Z{-1.8808) = 0.03 plyng X-75 $ So, y-75 8 -1.8808 ly ~ 60 59.9536 There fore , The lowest passing score is 6o.

Percentage Points for the Standard Normal distribution The table gives percentage points x defined by the equation 1 _/2t2 dt P= - vare 120 x 0 х Р х P x P x P x P x P x 50% 0.0000 5.0% 1.6449 3.0% 1.8808 2.0% 2.0537 1.0% 2.3263 45% 0.1257 4.8% 1.6646 2.9% 1.8957 1.9% 2.0749 0.9% 2.3656 40% 0.2533 4.6% 1.6849 2.8% 1.9110 1.8% 2.0969 0.8% 2.4089 35% 0.3853 4.4% 1.7060 2.7% 1.9268 1.7% 2.1201 0.7% 2.4573 30% 0.5244 4.2% 1.7279 2.6% 1.9431 1.6% 2.1444 0.6% 2.5121 0.10% 3.0902 0.09% 3.1214 0.08% 3.1559 0.07% 3.1947 0.06% 3.2389 25% 0.6745 4.0% 1.7507 2.5% 1.9600 1.5% 2.1701 0.5% 2.5758 20% 0.8416 3.8% 1.7744 2.4% 1.9774 1.4% 2.1973 0.4% 2.6521 15% 1.0364 3.6% 1.7991 2.3% 1.9954 1.3% 2.2262 0.3% 2.7478 10% 1.2816 3.4% 1.8250 2.2% 2.0141 1.2% 2.2571 0.2% 2.8782 5% 1.6449 3.2% 1.8522 2.1% 2.0335 1.1% 2.2904 0.1% 3.0902 0.05% 3.2905 0.01% 3.7190 0.005% 3.8906 0.001% 4.2649 0.0005% 4.4172