Questions: Vacation Planning About 1/4 of potential vacationers have delayed or cancelled a trip because of the stress of planning. If 400 potential vacation planners are randomly selected, find the mean, variance, and standard deviation of those who have delayed or cancelled a trip because of the stress of planning. Round your answers to at least three decimal places. Part: 0 / 2 Part 1 of 2 (a) Find the mean. Mean: μ=

Solution Steps

The mean \( \mu \) of a binomial distribution can be calculated using the formula:

\[ \mu = n \cdot p \]

where:

• \( n = 400 \) (the number of trials)
• \( p = \frac{1}{4} \) (the probability of success)

Substituting the values:

\[ \mu = 400 \cdot \frac{1}{4} = 100.0 \]

The variance \( \sigma^2 \) of a binomial distribution is given by the formula:

\[ \sigma^2 = n \cdot p \cdot q \]

where:

• \( q = 1 - p = \frac{3}{4} \)

Substituting the values:

\[ \sigma^2 = 400 \cdot \frac{1}{4} \cdot \frac{3}{4} = 75.0 \]

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

\[ \sigma = \sqrt{n \cdot p \cdot q} \]

Substituting the values:

\[ \sigma = \sqrt{400 \cdot \frac{1}{4} \cdot \frac{3}{4}} = \sqrt{75.0} \approx 8.66 \]

Final Answer