Question

Consider the variable x = time required for a college student to complete a standardized exam. Suppose that for the population of students at a particular university, the distribution of x is well approximated by a normal curve with mean 50 minutes and standard deviation 5 minutes. (You may need to use a table or technology. Round your answers to four decimal places.)

(a) If 55 minutes is allowed for the exam, what proportion of students at this university would be unable to finish in the allotted time?
(b) How much time (in minutes) should be allowed for the exam if you wanted 95% of the students taking the test to be able to finish in the allotted time? min
(c) How much time (in minutes) is required for the fastest 25% of all students to complete the exam? min

Answer 1

To find the proportion of students who would be unable to finish the exam in the allotted time, calculate the z-score for the given time limit and use a standard normal distribution table. To find the time required for 95% of the students to finish the exam, find the critical value from the standard normal distribution table. To find the time required for the fastest 25% of all students, find the z-score corresponding to a cumulative area of 0.25 in the standard normal distribution table.

Step-by-step explanation:

To find the proportion of students at this university who would be unable to finish the exam in the allotted time, we need to calculate the z-score for the given time limit of 55 minutes. The z-score formula is z = (x - mean) / standard deviation. Substituting the given values, we get z = (55 - 50) / 5 = 1. The proportion of students who would be unable to finish in the allotted time can be found using a standard normal distribution table or a calculator, and it corresponds to the area to the right of the z-score of 1.

To find the time required for 95% of the students to finish the exam, we need to find the z-score corresponding to a cumulative area of 95% in the standard normal distribution table. This z-score will be the critical value. Once we have the critical value, we can use it to find the corresponding time value using the z-score formula. Substituting the critical value and the given mean and standard deviation into the formula, we can solve for the time required.

To find the time required for the fastest 25% of all students to complete the exam, we need to find the z-score corresponding to a cumulative area of 0.25 in the standard normal distribution table. This z-score will be negative because we are looking for the fastest 25%. Once we have the z-score, we can use the z-score formula to find the corresponding time value.