Question

hw 9 - binomial distributions
score: 5.28/10 answered: 6/10
question 9
in a certain large metropolitan city, 18% of the women have ended their child - children. if 242 women who have ended their childbearing years are random of women who have not had any children, what are the values for the mean
μx=
σx=
(round to 2 decimal places)
what would be the expected number of women in this sample who have not had
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Explanation:

Step1: Identify distribution and parameters

This is a binomial distribution problem. Let \( n = 240 \) (number of trials), \( p = 0.19 \) (probability of success, here success is a woman having no children).

Step2: Calculate the mean (\(\mu_X\))

For a binomial distribution, the mean is given by \(\mu_X = np\).
Substitute \( n = 240 \) and \( p = 0.19 \):
\(\mu_X = 240\times0.19 = 45.6\)

Step3: Calculate the standard deviation (\(\sigma_X\))

The formula for the standard deviation of a binomial distribution is \(\sigma_X=\sqrt{np(1 - p)}\).
First, calculate \( 1-p=1 - 0.19 = 0.81 \).
Then, \( np(1 - p)=240\times0.19\times0.81 = 240\times0.1539 = 36.936 \).
Take the square root: \(\sigma_X=\sqrt{36.936}\approx6.08\) (rounded to 2 decimal places)

Answer:

\(\mu_X = 45.6\)
\(\sigma_X\approx6.08\)
The expected number of women (mean) is \( 45.6 \).