Answer to Question #218373 in Statistics and Probability for Leah

Question

a) The weekly wages of employees of Volta gold are normally distributed about a mean of $1,250 with a standard deviation of $120. Find the probability of an employee having a weekly wage lying;
i) Between $1,320 and $970
ii) Under $1,400
iii) Over $1,290

b) An achievement test was administered to a class of 20,000 students. The mean score was 80 and the standard deviation was 11. If Lingard scored 72 in the test, how many students did better than him?

Answer 1

a)

$\mu=1250 \\ \sigma=120 \\ i. \; P(970<X<1320) = P(X<1320) -P(X<970) \\ = P(Z< \frac{1320-1250}{120}) -P(Z< \frac{970-1250}{120}) \\ = P(Z< 0.5833) -P(Z< -2.333) \\ = 0.72005-0.00982 \\ = 0.71023 \\ ii. \;P(X<1400) = P(Z< \frac{1400-1250}{120}) \\ =P(Z< 1.25) \\ = 0.8943 \\ iii. \; P(X>1290) = 1 -P(X<1290) \\ = 1 -P(Z< \frac{1290-1250}{120}) \\ = 1 -P(Z<0.3333) \\ = 1 -0.63043 \\ = 0.36957$

b)

$N=20000 \\ \mu=80 \\ \sigma=11 \\ P(X>72) = 1 -P(X<72) \\ = 1 -P(Z< \frac{72-80}{11}) \\ = 1 -P(Z < -0.7272) \\ = 1 -0.2336 \\ = 0.7664 \\ n(X>72) = 20000 \times 0.7664 = 15328$

Answer: 15328 students were better than Lingard