Questions: In a survey, cell phone users were asked which ear they use to hear their cell phone, and the table is based on their responses. 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. P(x) Left 0.6355 Right 0.3041 No preference 0.0604

Solution Steps

To determine if a probability distribution is given, we need to check if the sum of all probabilities equals 1. If it is a valid probability distribution, we can then calculate the mean and standard deviation. The mean of a probability distribution is calculated as the sum of each value multiplied by its probability. The standard deviation is calculated using the formula for the standard deviation of a probability distribution.

To determine if the given probabilities form a probability distribution, we check if the sum of all probabilities equals 1. The given probabilities are:

• \( P(\text{Left}) = 0.6355 \)
• \( P(\text{Right}) = 0.3041 \)
• \( P(\text{No preference}) = 0.0604 \)

Calculate the sum:

\[ 0.6355 + 0.3041 + 0.0604 = 1.0000 \]

Since the sum is 1, the given probabilities form a valid probability distribution.

The mean \(\mu\) of a probability distribution is calculated as:

\[ \mu = \sum (x \cdot P(x)) \]

Assign numerical values to each category:

• Left: 1
• Right: 2
• No preference: 3

Calculate the mean:

\[ \mu = (1 \cdot 0.6355) + (2 \cdot 0.3041) + (3 \cdot 0.0604) = 1.4249 \]

The standard deviation \(\sigma\) is calculated using the formula:

\[ \sigma = \sqrt{\sum ((x - \mu)^2 \cdot P(x))} \]

Calculate the variance first:

\[ \begin{align_} \text{Variance} &= (1 - 1.4249)^2 \cdot 0.6355 + (2 - 1.4249)^2 \cdot 0.3041 + (3 - 1.4249)^2 \cdot 0.0604 \\ &= 0.1801 \end{align_} \]

Then, calculate the standard deviation:

\[ \sigma = \sqrt{0.1801} = 0.4244 \]

Final Answer