Questions: Refer to the accompanying table, which describes results from groups of 8 births from 6 different sets of parents. The random variable x represents the number of girls among 8 children. Find the mean and standard deviation for the number of girls in 8 births. The mean is μ = [ ] girl(s) (Round to one decimal place as needed.)

Solution Steps

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

$$\mu = n \cdot p$$

where:

• $n = 8$ (the number of trials),
• $p$ is the probability of success (having a girl).

From the calculations, we find:

$$\mu = 8 \cdot 0.5375 = 4.3$$

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

$$\sigma^2 = n \cdot p \cdot q$$

where $q = 1 - p$. Thus, we have:

$$\sigma^2 = 8 \cdot 0.5375 \cdot 0.4625 = 2.0$$

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

$$\sigma = \sqrt{n \cdot p \cdot q} = \sqrt{2.0} \approx 1.4$$

Final Answer