Question

Mean and Standard Deviation of Class Year In Exercise P.117, we discuss the random variable counting the number of seniors in a sample of four undergraduate students at a university, given that the proportion of undergraduate students who are seniors is \(0.25 .\) Find the mean and standard deviation of this random variable.

Short answer

The mean (expected value) of the random variable is 1. The standard deviation of the random variable is \(\sqrt{(4 \times 0.25 \times 0.75)}\).

Step-by-step solution

Identifying parameters of the binomial distribution

First identify the parameters for the binomial distribution problem. In this case, the number of trials, \(n\), is the sample size which is 4 (students) and the probability of success, \(p\), is the proportion of seniors which is 0.25.

Calculate the mean

Next, calculate the mean, also known as expected value, using the formula for a binomial distribution, which is \(n \times p\). Substituting the given values, the mean (\(\mu\)) is \(4 \times 0.25 = 1\).

Calculate the standard deviation

Now, calculate the standard deviation using its formula in the binomial distribution context, which is \(\sqrt{n \times p \times (1-p)}\). Substituting the given values, the standard deviation (\(\sigma\)) is \(\sqrt{(4 \times 0.25 \times 0.75)}\). Compute this to get the value of the standard deviation.