Questions: Based on data from a car bumper sticker study, when a car is randomly selected, the number of bumper stickers and the corresponding probabilities are as shown below. Complete parts (a) through (d). a. Does the given information describe a probability distribution? Yes No b. Assuming that a probability distribution is described, find its mean and standard deviation. The mean is (Round to the nearest tenth as needed.) The standard deviation is 1.4 (Round to the nearest tenth as needed.) c. Use the range rule of thumb to identify the range of values for usual numbers of bumper stickers. The maximum usual value is (Round to the nearest tenth as needed.) The minimum usual value is (Round to the nearest tenth as needed.)

Solution Steps

a. To determine if the given information describes a probability distribution, check if the sum of the probabilities is 1 and if all probabilities are between 0 and 1.

b. If it is a probability distribution, calculate the mean (expected value) using the formula \( \mu = \sum (x \cdot P(x)) \) and the standard deviation using the formula \( \sigma = \sqrt{\sum ((x - \mu)^2 \cdot P(x))} \).

c. Use the range rule of thumb to identify the range of usual values, which is given by \( \mu \pm 2\sigma \).

To determine if the given information describes a probability distribution, we check two conditions:

1. The sum of the probabilities must be 1.
2. All probabilities must be between 0 and 1.

Given: \[ \text{Probabilities} = [0.1, 0.2, 0.3, 0.2, 0.1, 0.1] \]

\[ \sum P(x) = 0.1 + 0.2 + 0.3 + 0.2 + 0.1 + 0.1 = 1 \]

Since both conditions are satisfied, the given information describes a probability distribution.

The mean \( \mu \) of a probability distribution is given by: \[ \mu = \sum (x \cdot P(x)) \]

Given: \[ \text{Bumper Stickers} = [0, 1, 2, 3, 4, 5] \] \[ \text{Probabilities} = [0.1, 0.2, 0.3, 0.2, 0.1, 0.1] \]

\[ \mu = 0 \cdot 0.1 + 1 \cdot 0.2 + 2 \cdot 0.3 + 3 \cdot 0.2 + 4 \cdot 0.1 + 5 \cdot 0.1 = 2.3 \]

The standard deviation \( \sigma \) is given by: \[ \sigma = \sqrt{\sum ((x - \mu)^2 \cdot P(x))} \]

\[ \sigma^2 = (0 - 2.3)^2 \cdot 0.1 + (1 - 2.3)^2 \cdot 0.2 + (2 - 2.3)^2 \cdot 0.3 + (3 - 2.3)^2 \cdot 0.2 + (4 - 2.3)^2 \cdot 0.1 + (5 - 2.3)^2 \cdot 0.1 \]

\[ \sigma^2 = 5.29 \cdot 0.1 + 1.69 \cdot 0.2 + 0.09 \cdot 0.3 + 0.49 \cdot 0.2 + 2.89 \cdot 0.1 + 7.29 \cdot 0.1 = 1.96 \]

\[ \sigma = \sqrt{1.96} = 1.4 \]

Using the range rule of thumb, the range of usual values is given by: \[ \text{Range} = \mu \pm 2\sigma \]

\[ \text{Maximum Usual Value} = \mu + 2\sigma = 2.3 + 2 \cdot 1.4 = 5.1 \]

\[ \text{Minimum Usual Value} = \mu - 2\sigma = 2.3 - 2 \cdot 1.4 = -0.5 \]

Final Answer