Question

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.
click the icon to view the data table.
the mean is μ = 4 girl(s). (round to one decimal place as needed.)
the standard deviation is σ = □ girl(s). (round to one decimal place as needed.)
table of numbers of girls and probabilities
number of
girls x
p(x)
0
0.004
1
0.032
2
0.112
3
0.212
4
0.280
5
0.222
6
0.108
7
0.027
8
0.003
print
done

Explanation:

Step1: Recall the formula for standard deviation of a discrete probability distribution

$$\sigma=\sqrt{\sum(x - \mu)^2P(x)}$$
where $\mu$ is the mean, $x$ is the value of the random variable, and $P(x)$ is the probability of $x$.

Step2: Calculate $(x-\mu)^2P(x)$ for each value of $x$

• For $x = 0$: $(0 - 4)^2\times0.004=16\times0.004 = 0.064$
• For $x = 1$: $(1 - 4)^2\times0.032=9\times0.032 = 0.288$
• For $x = 2$: $(2 - 4)^2\times0.112=4\times0.112 = 0.448$
• For $x = 3$: $(3 - 4)^2\times0.212=1\times0.212 = 0.212$
• For $x = 4$: $(4 - 4)^2\times0.280=0\times0.280 = 0$
• For $x = 5$: $(5 - 4)^2\times0.222=1\times0.222 = 0.222$
• For $x = 6$: $(6 - 4)^2\times0.108=4\times0.108 = 0.432$
• For $x = 7$: $(7 - 4)^2\times0.027=9\times0.027 = 0.243$
• For $x = 8$: $(8 - 4)^2\times0.003=16\times0.003 = 0.048$

Step3: Sum up the values

$$\sum(x - \mu)^2P(x)=0.064 + 0.288+0.448 + 0.212+0+0.222+0.432+0.243+0.048 = 1.959$$

Step4: Calculate the standard deviation

$$\sigma=\sqrt{1.959}\approx1.4$$

Answer:

The standard deviation is $\sigma = 1.4$ girl(s).