Question

Suppose that you are analyzing test scores of the final exam in your class that are normally distributed, which of the following expressions can be applied for a continuous probability density function when the mean of the test score is 79.5%?

a. The probability of obtaining a score of 79.5% on the final exam is 0.

b. The probability of obtaining a score of 79.5% on the final exam is 1.

c. The probability of obtaining a score greater than 79.5% on the final exam is 1.

Answer 1

Ans: a. The probability of obtaining a score of 79.5% on the final exam is 0

Reason: Value of a continuous probability density function (pdf) at a single particular value is zero. The reason is as follows - Suppose f(x) is a continuous probability density function defined over a to b (where a and b are real values). Now since it is continuous there are infinite number of points between the range a to b. Summation of probabilities at all of these points must be equal to 1 i.e. summation of values at infinite points must be equal to 1 hence the values at each point must be infinitesimal. That is the next best thing to actually being zero. Thus value at a perticupar point for a continuous pdf is zero.

Another way to look at it is that, consider the graph of pdf f(x) vs x. Probability will be the area under the curve (i.e. integration over a defined limit). When x is a single value (point) then area under the curve for that point is zero. Hence probability for a single value will be zero.

Hope this helped!