Question 3.11: Suppose that numerical grades in a statistics class are valu......
Suppose that numerical grades in a statistics class are values of a r.v. X which is (approximately) Normally distributed with mean μ = 65 and s.d. σ = 15. Furthermore, suppose that letter grades are assigned according to the following rule: the student receives an A if X ≥ 85; B if 70 ≤ X < 85; C if 55 ≤ X < 70; D if 45 ≤ X < 55; and F if X ≤ 45.
(i) If a student is chosen at random from that class, calculate the probability that the student will earn a given letter grade.
(ii) Identify the expected proportions of letter grades to be assigned.
(i) The student earns an A with probability P(X\geq85)=1-P(X\lt 85)= 1-P({\frac{X-\mu}{\sigma}}\lt {\frac{85-65}{15}})\simeq1-{{P}}(Z\lt 1.34)\simeq1-\Phi(1.34)=1-0.909877=0.090123 \simeq 0.09. Likewise, the student earns a B with probability P(70\leq X\lt 85)=P(\frac{70-65}{15}\le\frac{X-\mu}{\sigma}\lt\frac{85-65}{15})\simeq P(0.34\le Z\lt 1.34)\simeq\Phi(1.34)-\Phi(0.34)=0.909877-0.633072=0.276805\simeq 0.277 Similarly, the student earns a C with probability P(55\le X\lt 70)\simeq\Phi(0.34)+\Phi(0.67)-1=0.381643\simeq0.382. The student earns a D with probability P(45\le X\ \lt 55)\simeq\Phi(1.34)-\Phi(0.67)=0.161306\simeq0.161, and the student is assigned an F with probability P(X\lt 45)\simeq\Phi(-1.34)=1-\Phi(1.34)=0.09123\simeq 0.091
(ii) The respective expected proportions for A, B, C, D, and F are: 9%, 28%, 38%, 16%, and 9%.