A ski resort claims that there is a 75% chance of snow on any given day in January and that snowfall happens independently from one day to the next. A family plans a four-day trip to this ski resort in January. Let [tex]$X$[/tex] represent the number of days it snows while the family is there. What are the mean and standard deviation of [tex]$X$[/tex]?
A. [tex]$\mu_x=1, \sigma_x=1$[/tex]
B. [tex]$\mu_x=2, \sigma_x=0.56$[/tex]
C. [tex]$\mu_x=3, \sigma_x=0.75$[/tex]
D. [tex]$\mu_x=3, \sigma_x=0.87$[/tex]
Answer (2)
Recognize the problem as a binomial distribution with n = 4 and p = 0.75 .
Calculate the mean using the formula μ = n p = 4 × 0.75 = 3 .
Calculate the standard deviation using the formula σ = n p ( 1 − p ) = 4 × 0.75 × 0.25 = 0.75 ≈ 0.87 .
The mean and standard deviation are μ x = 3 , σ x = 0.87 .
Explanation
Understand the problem and provided data We are given that the probability of snow on any given day in January at a ski resort is 75%, or 0.75. A family plans a four-day trip to the resort. We need to find the mean and standard deviation of the number of days it snows during their trip. Let X be the number of days it snows.
Identify the distribution Since the snowfall on each day is independent, we can model this situation using a binomial distribution. The number of trials is n = 4 (the number of days of the trip), and the probability of success (snow) on each trial is p = 0.75 .
Calculate the mean The mean of a binomial distribution is given by the formula: E [ X ] = n p Substituting the values, we get: E [ X ] = 4 × 0.75 = 3
Calculate the variance The variance of a binomial distribution is given by the formula: Va r ( X ) = n p ( 1 − p ) Substituting the values, we get: Va r ( X ) = 4 × 0.75 × ( 1 − 0.75 ) = 4 × 0.75 × 0.25 = 3 × 0.25 = 0.75
Calculate the standard deviation The standard deviation is the square root of the variance: S D ( X ) = Va r ( X ) S D ( X ) = 0.75 ≈ 0.866
State the final answer Therefore, the mean number of days it snows is 3, and the standard deviation is approximately 0.87.
Examples
This type of problem is useful in risk assessment and planning. For example, a business might use binomial distribution to estimate the number of successful sales calls made in a week, given the probability of success for each call. Similarly, in weather forecasting, it can help estimate the number of rainy days in a month, which is crucial for agriculture and outdoor event planning. Understanding these probabilities allows for better resource allocation and decision-making.
The mean number of days it snows during the family's four-day trip to the ski resort is 3, while the standard deviation is approximately 0.87. Thus, the correct answer is option D: μ x = 3 , σ x = 0.87 .
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