Question

assume that different groups of couples use a particular method of gender selection and each couple gives birth to one baby. this method is designed to increase the likelihood that each baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5. assume that the groups consist of 31 couples. complete parts (a) through (c) below.
a. find the mean and the standard deviation for the numbers of girls in groups of 31 births.
the value of the mean is μ = 15.5
(type an integer or a decimal. do not round.)
the value of the standard deviation is σ = 2.8
(round to one decimal place as needed.)
b. use the range - rule of thumb to find the values separating results that are significantly low or significantly high.
values of 9.9 girls or lower are significantly low.
(round to one decimal place as needed.)
values of 21.1 girls or greater are significantly high.
(round to one decimal place as needed.)
c. is the result of 27 girls a result that is significantly high? what does it suggest about the effectiveness of the method?
the result significantly high, because 27 girls is girls. a result of 27 girls would suggest that the method

Explanation:

Step1: Recall the mean formula for binomial distribution

The mean $\mu$ of a binomial distribution is given by $\mu = np$, where $n$ is the number of trials and $p$ is the probability of success on each trial. Here, $n = 31$ (number of births) and $p=0.5$ (probability of a girl), so $\mu=np = 31\times0.5=15.5$.

Step2: Recall the standard - deviation formula for binomial distribution

The standard deviation $\sigma$ of a binomial distribution is $\sigma=\sqrt{np(1 - p)}$. Substitute $n = 31$ and $p = 0.5$: $\sigma=\sqrt{31\times0.5\times(1 - 0.5)}=\sqrt{31\times0.5\times0.5}=\sqrt{7.75}\approx2.8$.

Step3: Apply the range - rule of thumb for significant values

The range - rule of thumb for significant low values is $\mu - 2\sigma$ and for significant high values is $\mu+2\sigma$.
$\mu - 2\sigma=15.5-2\times2.8=15.5 - 5.6 = 9.9$.
$\mu+2\sigma=15.5 + 2\times2.8=15.5+5.6 = 21.1$.

Step4: Determine if 27 girls is significantly high

Since $27>21.1$, the result of 27 girls is significantly high. A significantly high result suggests that the method may be effective.

Answer:

The result is significantly high, because 27 girls is greater than 21.1 girls. A result of 27 girls would suggest that the method is effective.