Question: The average one-way commuting time in Richmond, Virginia is 21.9 minutes with a standard deviation…

The average one-way commuting time in Richmond, Virginia is 21.9 minutes with a standard deviation of 8.6 minutes. If commuting times are normally distributed, what is the probability that the average of 8 commuters in one day is less than 20 minutes?

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Numeric 6 points

The probability that a U.S. household has a particular number of smartphones is given in the following table:

Phones P 0 .152 1 .202 2 .313 3 .333

What is the average number of smartphones in a U.S. household?

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Solution

To find the average number of smartphones in a U.S. household, calculate the expected value using the given probabilities and number of phones. Let \( X \) be the random variable representing the number of smartphones. Then, the probabilities are given as follows: \[ \begin{cases} P(X = 0) = 0.152 & \\ P(X = 1) = 0.202 & \\ P(X = 2) = 0.313 & \\ P(X = 3) = 0.333 & \end{cases} \] The expected value \( E(X) \) is calculated as: \[ E(X) = (0 \times 0.152) + (1 \times 0.202) + (2 \times 0.313) + (3 \times 0.333) \] Calculate each term: \[ 0 \times 0.152 = 0 \] \[ 1 \times 0.202 = 0.202 \] \[ 2 \times 0.313 = 0.626 \] \[ 3 \times 0.333 = 0.999 \] Add these values: \[ E(X) = 0 + 0.202 + 0.626 + 0.999 = 1.827 \] Therefore, the average number of smartphones in a U.S. household is 1.827.