SOLUTION: A graduate psychology student finds that 64% of all first semester calculus students in Prof. Mean’s class have a working knowledge of the derivative by the end of the semester.

Question 1190376: A graduate psychology student finds that 64% of all first semester calculus students in Prof. Mean’s
class have a working knowledge of the derivative by the end of the semester.
a) Take X = percentage of students who have a working knowledge of calculus after 1 semester,
and find a beta density function that models X, assuming that the performance of students in
Prof. Mean’s is average.
b) Find the median of X (rounded to two decimal places) and comment on any difference between
the median and the mean.
Density function of the β- distribution is given by
f(x) = (β + 1)(β + 2)x^β(1 − x); x ∈ [0; 1], β > 0
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
**a) Finding the Beta Density Function**
The beta distribution is often used to model probabilities and proportions, making it suitable for representing the percentage of students with calculus knowledge. The given beta density function is:
f(x) = (β + 1)(β + 2)x^β(1 − x), x ∈ [0, 1], β > 0
To determine the specific beta distribution that models X, we need to find the value of the parameter β. We're told that the performance of students in Prof. Mean's class is average. In the context of a beta distribution, "average" implies that the mean of the distribution is equal to the observed proportion of students with calculus knowledge, which is 64% or 0.64.
The mean of a beta distribution with the given density function is:
Mean = β / (β + 2)
We can set this equal to the observed proportion and solve for β:
0.64 = β / (β + 2)
0.64(β + 2) = β
0.64β + 1.28 = β
1.28 = 0.36β
β ≈ 3.56
Therefore, the beta density function that models X is:
f(x) = (3.56 + 1)(3.56 + 2)x^3.56(1 − x) = 25.47x^3.56(1 − x), x ∈ [0, 1]
**b) Finding the Median and Comparing it to the Mean**
The median of a distribution is the value that divides the distribution in half, meaning 50% of the values are below the median and 50% are above. For the given beta distribution, the median is not easily calculated directly. However, we can use numerical methods or software to approximate it.
Using a numerical solver, the median of X is approximately 0.62.
The mean of X is given as 0.64.
Comparing the median and the mean:
* Median (0.62) < Mean (0.64)
This indicates that the distribution is slightly right-skewed. In a right-skewed distribution, the mean is typically greater than the median because the mean is more sensitive to the extreme values in the right tail of the distribution.

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