Question

college students are randomly selected and arranged in groups of three. the random variable x is the number in the group who say that they take one or more online courses. determine whether a probability distribution is given. if a probability distribution is given, find its mean and standard deviation. if a probability distribution is not given, identify the requirements that are not satisfied.
does the table show a probability distribution? select all that apply
a. yes, the table shows a probability distribution.
b. no, the random variable x is categorical instead of numerical.
c. no, the numerical values of the random variable x are not associated with probabilities.
d. no, not every probability is between 0 and 1 inclusive.
e. no, the sum of all the probabilities is not equal to 1.
find the mean of the random variable x. select the correct choice below and, if necessary, fill in the answer box to complete your choice
a. μ = students (round to one decimal place as needed )
b. the table does not show a probability distribution.

Explanation:

Step1: Check probability - distribution requirements

1. All probabilities \(P(x)\): \(0.103\), \(0.351\), \(0.399\), \(0.147\) are between \(0\) and \(1\) inclusive.
2. Sum of probabilities: \(0.103 + 0.351+0.399 + 0.147=1\). Also, \(x = 0,1,2,3\) is a numerical random - variable with associated probabilities. So, it is a probability distribution.

Step2: Calculate the mean \(\mu\)

The formula for the mean of a discrete probability distribution is \(\mu=\sum x\cdot P(x)\).
\[

$$\begin{align*} \mu&=(0\times0.103)+(1\times0.351)+(2\times0.399)+(3\times0.147)\\ &=0 + 0.351+0.798 + 0.441\\ &=1.6 \end{align*}$$

\]

Answer:

A. Yes, the table shows a probability distribution
A. \(\mu = 1.6\) students