In a family of five children, there is a [tex]$25 \%$[/tex] chance that each individual child has freckles. Children inherit independently of one another. Let [tex]$X$[/tex] represent the number of children in this family with freckles. What are the mean and standard deviation of [tex]$X$[/tex]?
A. [tex]$\mu_x=1.25, \sigma_x=0.97$[/tex]
B. [tex]$\mu_x=1.25, \sigma_x=0.94$[/tex]
C. [tex]$\mu_x=1.5, \sigma_x=0.19$[/tex]
D. [tex]$\mu_x=2.5, \sigma_x=0.97$[/tex]
Answer (2)
Calculate the mean: μ x = 5 ∗ 0.25 = 1.25 .
Calculate the variance: σ x 2 = 5 ∗ 0.25 ∗ 0.75 = 0.9375 .
Calculate the standard deviation: σ x = 0.9375 ≈ 0.97 .
The mean and standard deviation are μ x = 1.25 , σ x = 0.97 .
Explanation
Understand the problem We are given a problem where we need to find the mean and standard deviation of the number of children with freckles in a family of five, where each child has a 25% chance of having freckles. The children inherit freckles independently. This scenario can be modeled using a binomial distribution.
Calculate the mean In a binomial distribution, the mean ( μ x ) is calculated as n ∗ p , where n is the number of trials (in this case, the number of children) and p is the probability of success (in this case, the probability of a child having freckles). So, we have: μ x = n ∗ p = 5 ∗ 0.25 = 1.25 Thus, the mean number of children with freckles is 1.25.
Calculate the variance The variance ( σ x 2 ) of a binomial distribution is calculated as n ∗ p ∗ ( 1 − p ) . Therefore: σ x 2 = n ∗ p ∗ ( 1 − p ) = 5 ∗ 0.25 ∗ ( 1 − 0.25 ) = 5 ∗ 0.25 ∗ 0.75 = 0.9375
Calculate the standard deviation The standard deviation ( σ x ) is the square root of the variance. So, σ x = σ x 2 = 0.9375 ≈ 0.9682 Rounding to two decimal places, the standard deviation is approximately 0.97.
State the final answer Therefore, the mean number of children with freckles is 1.25, and the standard deviation is approximately 0.97.
Examples
Consider a scenario where a genetic trait appears with a certain probability in a population. For instance, if we are studying a family with a history of a particular genetic marker, we can use the binomial distribution to estimate the expected number of family members who will express the trait and the variability around that expectation. This helps in genetic counseling and understanding the inheritance patterns of the trait within the family.
The mean number of children with freckles in the family is 1.25, while the standard deviation is approximately 0.97. This is calculated using a binomial distribution model where each child has a 25% chance of having freckles. Therefore, the answer is option A: μ x = 1.25 , σ x = 0.97 .
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