Question 5, 5.1.12
Part 3 of 3
4 correct
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Groups of adults are randomly selected and arranged in groups of three. The random variable $x$ is the number in the group who say that they would feel comfortable in a self-driving vehicle. 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.
\begin{tabular}{c|c}
\hline$x$ & $P(x)$ \\
\hline 0 & 0.353 \\
\hline 1 & 0.426 \\
\hline 2 & 0.200 \\
\hline 3 & 0.021 \\
\hline
\end{tabular}
Does the table show a probability distribution? Select all that apply.
A. Yes, the table shows a probability distribution.
B. No, the random variable $x$ 's number values are not associated with probabilities.
C. No, the random variable $\mathrm{x}$ is categorical instead of numerical.
D. No, the sum of all the probabilities is not equal to 1 .
E. No, not every probability is between 0 and 1 inclusive.
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. $\mu=0.9$ adult(s) (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. $\sigma=\square$ adult(s) (Round to one decimal place as needed.)
B. The table does not show a probability distribution.
Answer
The mean of the random variable x is \(\boxed{0.9}\) adult(s). The standard deviation of the random variable x is \(\boxed{0.9}\) adult(s).
Step 1 ：First, we need to check if the given table represents a probability distribution. For a table to represent a probability distribution, it must satisfy two conditions: 1. All the probabilities must be between 0 and 1, inclusive. 2. The sum of all the probabilities must be equal to 1. If the table satisfies these conditions, then it represents a probability distribution. If it does not, then it does not represent a probability distribution.
Step 2 ：If the table represents a probability distribution, we can then calculate the mean and standard deviation. The mean of a probability distribution is calculated by multiplying each possible outcome by its probability and then summing these products. The standard deviation is a measure of the dispersion of the probability distribution and is calculated using the formula: \[\sigma = \sqrt{\sum (x - \mu)^2 * P(x)}\] where: - $x$ is a possible outcome, - $\mu$ is the mean of the distribution, - $P(x)$ is the probability of $x$.
Step 3 ：The given table represents a probability distribution. The mean is 0.9 and the standard deviation is 0.9.
Step 4 ：The mean of the random variable x is \(\boxed{0.9}\) adult(s). The standard deviation of the random variable x is \(\boxed{0.9}\) adult(s).
