To determine whether 1 girl in 10 births is a significantly low number of girls using the range rule of thumb, we can follow these steps:
1. Calculate the Mean (Expected Value):
The mean [tex]\(\mu\)[/tex] is calculated as follows:
[tex]\[
\mu = \sum (x \cdot P(x))
\][/tex]
Given the results, the mean number of girls is [tex]\(4.986\)[/tex].
2. Calculate the Variance:
The variance [tex]\(\sigma^2\)[/tex] is calculated by:
[tex]\[
\sigma^2 = \sum ((x - \mu)^2 \cdot P(x))
\][/tex]
The variance is [tex]\(2.893804\)[/tex].
3. Calculate the Standard Deviation:
The standard deviation [tex]\(\sigma\)[/tex] is the square root of the variance:
[tex]\[
\sigma = \sqrt{2.893804} \approx 1.7011184556050174
\][/tex]
4. Determine the Range Using the Range Rule of Thumb:
The range rule of thumb states that usual values lie within 2 standard deviations of the mean:
[tex]\[
\text{Usual Minimum} = \mu - 2\sigma
\][/tex]
[tex]\[
\text{Usual Maximum} = \mu + 2\sigma
\][/tex]
Plugging in the values:
[tex]\[
\text{Usual Minimum} = 4.986 - 2 \times 1.7011184556050174 \approx 1.583763088789965
\][/tex]
[tex]\[
\text{Usual Maximum} = 4.986 + 2 \times 1.7011184556050174 \approx 8.388236911210035
\][/tex]
5. Assess if 1 Girl is Significantly Low:
To determine if 1 girl in 10 births is significantly low, check if it falls below the usual minimum value:
[tex]\[
1 < 1.583763088789965
\][/tex]
Since 1 girl is less than the usual minimum value of approximately 1.6, it is indeed a significantly low number of girls among 10 children.
To summarize, using the range rule of thumb:
- The maximum usual value is approximately [tex]\(8.4\)[/tex] girls.
- 1 girl in 10 births is considered a significantly low number of girls.
