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, not every probability is between 0 and 1 inclusive.
c. no, the sum of all the probabilities is not equal to 1.
d. no, the numerical values of the random variable x are not associated with probabilities.
e. no, the random variable x is categorical instead of numerical.

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 = 1.6 students
(round to one decimal place as needed.)
b. the table does not show a probability distribution.

find the standard deviation of the random variable x. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. σ = students
(round to one decimal place as needed.)
b. the table does not show a probability distribution.

Explanation:

Step1: Check if it is a probability distribution

For a table to be a probability distribution:

1. All \(P(x)\geq0\). Here, \(P(0) = 0.108\), \(P(1)=0.357\), \(P(2)=0.388\), \(P(3)=0.147\) are all non - negative.
2. \(\sum P(x)=0.108 + 0.357+0.388 + 0.147=1\)

Since both conditions are met, it is a probability distribution.

Step2: Calculate the mean \(\mu\)

The formula for the mean of a discrete probability distribution is \(\mu=\sum xP(x)\)

$$
LATEXBLOCK0
$$

Step3: Calculate the variance \(\sigma^{2}\)

The formula for the variance \(\sigma^{2}=\sum(x-\mu)^{2}P(x)\)

$$
LATEXBLOCK1
$$

\(\sigma^{2}=0.27648+0.12852 + 0.06208+0.28812=0.7552\)

Step4: Calculate the standard deviation \(\sigma\)

The formula for the standard deviation \(\sigma=\sqrt{\sigma^{2}}\)
\(\sigma=\sqrt{0.7552}\approx0.9\)

Answer:

A. Yes, the table shows a probability distribution.

A. \(\mu = 1.6\) students

A. \(\sigma=0.9\) students