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 45 minutes and standard deviation 5 minutes. a. If 50 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 should be allowed for the exam if you wanted \(90 \%\) of the students taking the test to be able to finish in the allotted time? c. How much time is required for the fastest \(25 \%\) of all tudents to complete the exam

Short answer

a. The proportion of students who would be unable to finish the exam in 50 minutes can be calculated using the Z-score and the standard normal distribution table. b. The time that should be allowed for the exam, given we want \(90\%\) of all students to be able to finish in time can be calculated after determining the corresponding Z-score for \(90\%\), and using it in the formula \(X = Z\sigma + \mu\). c. To find the time required for the fastest \(25\%\) of students to complete the exam, get the Z-score for \(25\%\), and compute using same formula \(X = Z\sigma + \mu\).

Step-by-step solution

Determine the Proportion Who Would Be Unable to Finish on Time

To find the proportion of students who would be unable to finish in 50 minutes, we need to calculate the z-score for 50 minutes using formula \(Z = (X - \mu) / \sigma \), where \(X=50\), \(\mu=45\) minutes and \(\sigma=5\). After calculating the Z-score, we can use the standard normal distribution table to find the proportion. This will give us the probability that a randomly chosen student will finish the exam in 50 minutes. However, we want to find the proportion of students who would be unable to finish in this time, therefore we need to subtract the calculated probability from 1.

Determine the Time That Should Be Allowed for Exam

To find out how much time should be allowed for the exam if you wanted \(90\%\) of the students taking the test to be able to finish in the allotted time, we need to find the Z-score corresponding to \(90\%\) from the standard normal distribution table. Then we can use formula \(X = Z\sigma + \mu\) to calculate the time, where Z is the Z-score found from the table, \(\mu = 45\) and \(\sigma = 5\).

Determine the Time Required for the Fastest Students

To find the time required for the fastest \(25\%\) of students to complete the exam, we first need to find the Z-score corresponding to \(25\%\) from the standard normal distribution table, then again using the formula \(X = Z\sigma + \mu\), we can calculate the time needed.