Answer to Question #145939 in Statistics and Probability for Vladimyr Lubin

Question

(5.1.16) The accompanying table describes results from groups of 10 births from 10 different sets of parents. The random variable x represents the number of girls among 10 children. Use the range rule of thumb to determine whether 1 girl in 10 births is a significantly low number of girls.
Number of
Girls x P(x)
0 0.003
1 0.016
2 0.036
3 0.112
4 0.208
5 0.233
6 0.199
7 0.115
8 0.042
9 0.012
10 0.024
Use the range rule of thumb to identify a range of values that are not significant.
The maximum value in this range is ?? girls.

Answer 1

The range rule of thumb tells that the range is generally about four times the standard

deviation.

$\mu=E(X)=\sum xP(x)$

$E(X)=0\cdot0.003+1\cdot0.016+2\cdot0.036+3\cdot0.112+4\cdot0.208+5\cdot0.233+6\cdot0.199+7\cdot0.115+8\cdot0.042+9\cdot0.012+10\cdot0.024=5.104$

Standard deviation:

$\sigma=SD(X)=\sqrt{E(X^2)-[E(X)]^2}$

$E(X^2)=\sum x^2P(x)$

$E(X^2)=0^2\cdot0.003+1^2\cdot0.016+2^2\cdot0.036+3^2\cdot0.112+4^2\cdot0.208+5^2\cdot0.233+6^2\cdot0.199+7^2\cdot0.115+8^2\cdot0.042+9^2\cdot0.012+10^2\cdot0.024=29.18$

$\sigma=\sqrt{29.18-5.104^2}=1.769$

The Range Rule of Thumb says that the range is about four times the standard deviation.

$$R=4\sigma$$

Since range $R=Max\ value - Min\ value$ lets determine the maximum value as:

$Max \ value = \mu+2\sigma=5.104+2\cdot1.769=8.642$

And use the range rule of thumb to determine the minimum value as:

$Min \ value = \mu-2\sigma=5.104-2\cdot1.769=1.566$

Therefore, the number of girls among the birth of the 10 children should fall between the range 1.566 and 8.642 i.e. taking the whole number of girls should fall between the range 2 and 9.

Hence, yes 1 girl in 10 births is a significantly low number of girls.

A range of values that are not significant is 0 to 2.

The maximum value in this range of girls is 9.