A sample of 500 evening students revealed that their annual incomes from employment in industry during the day were normally distributed with a mean income of $30,000 and a standard deviation of $3,000. (i) 250 students earned more than $30,000.
(ii. 500 students earned between $20,000 and $40,000.
(iii) 11 students earned more than $36,000.
A) (i), (ii), and (iii) are all correct statements.
B) (i) is a correct statement but not (ii) or (iii).
C) (i) and (iii) are correct statements but not (ii).
D) (i) and (ii) are correct statements but not (iii).
E) (ii) and (iii) are correct statements but not (i).

Question

Quizplus · Accepted Answer

(i) Since the mean income is $30,000, half of the sample (250) earn more than $30,000.
(ii) To calculate the proportion of students earning between $20,000 and $40,000, we need to standardize these values using z-scores. 
Calculating the z-scores for $20,000 and $40,000: 
z($20,000) = ($20,000 - $30,000) / $3,000 = -3.33 
z($40,000) = ($40,000 - $30,000) / $3,000 = 3.33 
Using a standard normal table or calculator, we can find that the proportion of values between -3.33 and 3.33 is equal to 0.9986. This means that nearly all (99.86%) of the sample falls between $20,000 and $40,000. 
(iii) We need to calculate the proportion of students earning more than $36,000 using a z-score:
z($36,000) = ($36,000 - $30,000) / $3,000 = 2 
Using a standard normal table or calculator, we can find that the proportion of values greater than 2 is 0.0228, or about 2.28%. This means that about 11 (0.0228 x 500) students in the sample earned more than $36,000.

A Sample of 500 Evening Students Revealed That Their Annual