Questions: Coffee with Meals A researcher wishes to determine the number of cups of coffee a customer drinks with an evening meal restaurant. X 0 1 2 3 4 P(X) 0.21 0.30 0.46 0.01 0.02 Find the mean. Round the answer to two decimal places as needed. Mean: μ=1.33 Find the variance. Round the answer to two decimal places as needed. Variance: σ^2=

Solution Steps

The mean \( \mu \) of the distribution is calculated using the formula:

\[ \mu = \sum_{i=0}^{n} x_i \cdot P(x_i) \]

Substituting the values:

\[ \mu = 0 \times 0.21 + 1 \times 0.30 + 2 \times 0.46 + 3 \times 0.01 + 4 \times 0.02 = 1.33 \]

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \sum_{i=0}^{n} (x_i - \mu)^2 \cdot P(x_i) \]

Substituting the values:

\[ \sigma^2 = (0 - 1.33)^2 \times 0.21 + (1 - 1.33)^2 \times 0.3 + (2 - 1.33)^2 \times 0.46 + (3 - 1.33)^2 \times 0.01 + (4 - 1.33)^2 \times 0.02 = 0.78 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{0.78} \approx 0.88 \]

Final Answer