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Refer to the accompanying table, which describes results from groups of 8 births from 8 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.
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The mean is $\mu=\square$ girl(s). (Round to one decimal place as needed.)
The standard deviation is $\sigma=\square$ girl(s). (Round to one decimal place as needed.)
Table of numbers of girls and probabilities
\begin{tabular}{c|c|}
\begin{tabular}{c}
Number of \\
Girls $x$
\end{tabular} & $P(x)$ \\
\hline 0 & 0.005 \\
1 & 0.031 \\
2 & 0.108 \\
3 & 0.213 \\
4 & 0.277 \\
5 & 0.219 \\
6 & 0.113 \\
7 & 0.029 \\
8 & 0.005 \\
\hline
\end{tabular}

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Step 1 ：Given a table of the number of girls and their probabilities, we can calculate the mean and standard deviation.

Step 2 ：The mean is calculated by summing the product of each number of girls and its corresponding probability. In mathematical terms, this is represented as \( \mu = \sum xP(x) \).

Step 3 ：Substituting the given values into the formula, we get \( \mu = (0 \times 0.005) + (1 \times 0.031) + (2 \times 0.108) + (3 \times 0.213) + (4 \times 0.277) + (5 \times 0.219) + (6 \times 0.113) + (7 \times 0.029) + (8 \times 0.005) = 4.01 \).

Step 4 ：We round this to one decimal place to get \( \mu = \boxed{4.0} \) girls.

Step 5 ：The variance is calculated by summing the product of the square of the difference between each number of girls and the mean, and its corresponding probability. In mathematical terms, this is represented as \( \sigma^2 = \sum (x - \mu)^2P(x) \).

Step 6 ：Substituting the given values into the formula, we get \( \sigma^2 = ((0 - 4.01)^2 \times 0.005) + ((1 - 4.01)^2 \times 0.031) + ((2 - 4.01)^2 \times 0.108) + ((3 - 4.01)^2 \times 0.213) + ((4 - 4.01)^2 \times 0.277) + ((5 - 4.01)^2 \times 0.219) + ((6 - 4.01)^2 \times 0.113) + ((7 - 4.01)^2 \times 0.029) + ((8 - 4.01)^2 \times 0.005) = 2.0159 \).

Step 7 ：The standard deviation is the square root of the variance. So, \( \sigma = \sqrt{2.0159} = 1.4198 \).

Step 8 ：We round this to one decimal place to get \( \sigma = \boxed{1.4} \) girls.