Questions: The professor of an introductory calculus class has stated that, historically, the distribution of final exam grades in the course resembles a normal distribution with a mean final exam mark of μ=61% and a standard deviation of σ=12%. (c) The top 3% of students writing the final exam will receive a letter grade of at least A in the course. To two decimal places, find the minimum final exam mark needed to earn a letter grade of at least A in the course.

Solution Steps

To find the minimum final exam mark needed to earn a letter grade of at least A, we first need to determine the Z-score that corresponds to the top \(3\%\) of students. This is equivalent to finding the Z-score for the \(97\)th percentile of the normal distribution.

Using the Z-score formula: \[ z = \frac{X - \mu}{\sigma} \] where \(X\) is the value we want to find, \(\mu = 0.5\) (mean of the standard normal distribution), and \(\sigma = 1\) (standard deviation of the standard normal distribution), we find: \[ z = 0.47 \]

Next, we use the Z-score to find the corresponding final exam mark using the formula: \[ X = \mu + z \cdot \sigma \] Substituting the known values: \[ X = 61 + 0.47 \cdot 12 \] Calculating this gives: \[ X = 61 + 5.64 = 66.64 \]

Final Answer