Math / Data & Statistics

QuestionThe 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 $\mathbf{x}$ & $\mathbf{P}(\mathbf{x})$ \\ \hline 0 & 0.362 \\ \hline 1 & 0.443 \\ \hline 2 & 0.172 \\ \hline 3 & 0.023 \\ \hline \end{tabular}
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 random variable x's number values are not associated with probabilities. D. No, the sum of all the probabilities is not equal to 1 . 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. $\mu=$ $\square$ adult(s) (Round to one decimal place as needed.) B. The table does not show a probability distribution.

Studdy Solution

STEP 1

What is this asking? We're checking if a table represents a valid probability distribution, and if it does, we need to calculate its mean and standard deviation. Watch out! A probability distribution must have probabilities between 0 and 1 that add up to 1, and the variable must be numerical.

STEP 2

1. Check if it's a probability distribution
2. Calculate the mean
3. Calculate the standard deviation

STEP 3

Let's look at the given probabilities: $0.362$ , $0.443$ , $0.172$ , and $0.023$ .
Are they all between $0$ and $1$ ?
Yes! Awesome!

STEP 4

Now, let's add them all up: $0.362 + 0.443 + 0.172 + 0.023 = 1$ .
Perfect! They add up to $1$ , just like they should for a probability distribution.

STEP 5

The random variable $x$ represents the number of adults in a group who are comfortable with self-driving vehicles.
This is a numerical value, so we're good here.

STEP 6

Since all the conditions are met, we can conclude that the table does show a probability distribution.

STEP 7

Remember, the mean ( $\mu$ ) of a probability distribution is calculated as: $\mu = \sum x \cdot P(x)$

STEP 8

Let's apply this formula to our table: $\mu = (0 \cdot 0.362) + (1 \cdot 0.443) + (2 \cdot 0.172) + (3 \cdot 0.023)$ $\mu = 0 + 0.443 + 0.344 + 0.069$ $\mu = 0.856$

STEP 9

So, the mean is $0.856$ .
This means that in a random group of three adults, we expect, on average, about $0.856$ of them to be comfortable with self-driving vehicles.

STEP 10

The standard deviation ( $\sigma$ ) is calculated as: $\sigma = \sqrt{\sum (x - \mu)^2 \cdot P(x)}$

STEP 11

Using our calculated mean ( $\mu = 0.856$ ): $\sigma = \sqrt{(0 - 0.856)^2 \cdot 0.362 + (1 - 0.856)^2 \cdot 0.443 + (2 - 0.856)^2 \cdot 0.172 + (3 - 0.856)^2 \cdot 0.023}$ $\sigma = \sqrt{0.265 + 0.009 + 0.213 + 0.101}$ $\sigma = \sqrt{0.588}$ $\sigma \approx 0.767$

STEP 12

The standard deviation is approximately $0.767$ .
This tells us how spread out the data is around the mean.

STEP 13

The table does represent a probability distribution (A).
The mean is $0.856$ adults (A).
The standard deviation is approximately $0.767$ .

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Studdy Solution

STEP 1

What is this asking? We're checking if a table represents a valid probability distribution, and if it does, we need to calculate its mean and standard deviation. Watch out! A probability distribution must have probabilities between 0 and 1 that add up to 1, and the variable must be numerical.

STEP 2

1. Check if it's a probability distribution
2. Calculate the mean
3. Calculate the standard deviation

STEP 3

Let's look at the given probabilities: $0.362$ , $0.443$ , $0.172$ , and $0.023$ .
Are they all between $0$ and $1$ ?
Yes! Awesome!

STEP 4

Now, let's add them all up: $0.362 + 0.443 + 0.172 + 0.023 = 1$ .
Perfect! They add up to $1$ , just like they should for a probability distribution.

STEP 5

The random variable $x$ represents the number of adults in a group who are comfortable with self-driving vehicles.
This is a numerical value, so we're good here.

STEP 6

Since all the conditions are met, we can conclude that the table does show a probability distribution.

STEP 7

Remember, the mean ( $\mu$ ) of a probability distribution is calculated as: $\mu = \sum x \cdot P(x)$

STEP 8

Let's apply this formula to our table: $\mu = (0 \cdot 0.362) + (1 \cdot 0.443) + (2 \cdot 0.172) + (3 \cdot 0.023)$ $\mu = 0 + 0.443 + 0.344 + 0.069$ $\mu = 0.856$

STEP 9

So, the mean is $0.856$ .
This means that in a random group of three adults, we expect, on average, about $0.856$ of them to be comfortable with self-driving vehicles.

STEP 10

The standard deviation ( $\sigma$ ) is calculated as: $\sigma = \sqrt{\sum (x - \mu)^2 \cdot P(x)}$

STEP 11

Using our calculated mean ( $\mu = 0.856$ ): $\sigma = \sqrt{(0 - 0.856)^2 \cdot 0.362 + (1 - 0.856)^2 \cdot 0.443 + (2 - 0.856)^2 \cdot 0.172 + (3 - 0.856)^2 \cdot 0.023}$ $\sigma = \sqrt{0.265 + 0.009 + 0.213 + 0.101}$ $\sigma = \sqrt{0.588}$ $\sigma \approx 0.767$

STEP 12

The standard deviation is approximately $0.767$ .
This tells us how spread out the data is around the mean.

STEP 13

The table does represent a probability distribution (A).
The mean is $0.856$ adults (A).
The standard deviation is approximately $0.767$ .

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Studdy Solution

STEP 1

What is this asking? We're checking if a table represents a valid probability distribution, and if it does, we need to calculate its mean and standard deviation. Watch out! A probability distribution must have probabilities between 0 and 1 that add up to 1, and the variable must be numerical.

STEP 2

1. Check if it's a probability distribution2. Calculate the mean3. Calculate the standard deviation

STEP 3

Let's **look** at the given probabilities: 0.3620.3620.362, 0.4430.4430.443, 0.1720.1720.172, and 0.0230.0230.023.Are they all between 000 and 111?Yes! Awesome!

STEP 4

Now, let's **add** them all up: 0.362+0.443+0.172+0.023=10.362 + 0.443 + 0.172 + 0.023 = 10.362+0.443+0.172+0.023=1.Perfect! They add up to 111, just like they should for a probability distribution.

STEP 5

The random variable xxx represents the number of adults in a group who are comfortable with self-driving vehicles.This is a numerical value, so we're good here.

STEP 6

Since all the conditions are met, we can **conclude** that the table *does* show a probability distribution.

STEP 7

Remember, the mean (μ\muμ) of a probability distribution is calculated as: μ=∑x⋅P(x) \mu = \sum x \cdot P(x) μ=∑x⋅P(x)

STEP 8

STEP 9

So, the **mean** is 0.8560.8560.856.This means that in a random group of three adults, we expect, on average, about 0.8560.8560.856 of them to be comfortable with self-driving vehicles.

STEP 10

The standard deviation (σ\sigmaσ) is calculated as: σ=∑(x−μ)2⋅P(x) \sigma = \sqrt{\sum (x - \mu)^2 \cdot P(x)} σ=∑(x−μ)2⋅P(x)​

STEP 11

STEP 12

The **standard deviation** is approximately 0.7670.7670.767.This tells us how spread out the data is around the mean.

STEP 13

The table *does* represent a probability distribution (A).The mean is 0.8560.8560.856 adults (A).The standard deviation is approximately 0.7670.7670.767.

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Check if it's a probability distribution
2. Calculate the mean
3. Calculate the standard deviation

Let's look at the given probabilities: $0.362$ , $0.443$ , $0.172$ , and $0.023$ .
Are they all between $0$ and $1$ ?
Yes! Awesome!

Now, let's add them all up: $0.362 + 0.443 + 0.172 + 0.023 = 1$ .
Perfect! They add up to $1$ , just like they should for a probability distribution.

The random variable $x$ represents the number of adults in a group who are comfortable with self-driving vehicles.
This is a numerical value, so we're good here.

Since all the conditions are met, we can conclude that the table does show a probability distribution.

Remember, the mean ( $\mu$ ) of a probability distribution is calculated as: $\mu = \sum x \cdot P(x)$

So, the mean is $0.856$ .
This means that in a random group of three adults, we expect, on average, about $0.856$ of them to be comfortable with self-driving vehicles.

The standard deviation ( $\sigma$ ) is calculated as: $\sigma = \sqrt{\sum (x - \mu)^2 \cdot P(x)}$

The standard deviation is approximately $0.767$ .
This tells us how spread out the data is around the mean.

The table does represent a probability distribution (A).
The mean is $0.856$ adults (A).
The standard deviation is approximately $0.767$ .