Questions: In a family with 2 children, the probability that all the children are girls is approximately 0.25 . In a random sample of 1000 families with 2 children, what is. the approximate probability that 220 or fewer will have 2 girls? Approximate a binomial distribution with a normal distribution. Use the table of areas under the normal curve given below. Click here to view page 1. Click here to view page 2. Click here to view page 3. Click here to view page 4. Click here to view page 5. Click here to view page 6.

Solution Steps

For a binomial distribution with parameters \( n = 1000 \) and \( p = 0.25 \):

• Mean \( \mu \) is calculated as: \[ \mu = n \cdot p = 1000 \cdot 0.25 = 250.0 \]

• Variance \( \sigma^2 \) is calculated as: \[ \sigma^2 = n \cdot p \cdot q = 1000 \cdot 0.25 \cdot 0.75 = 187.5 \]

• Standard Deviation \( \sigma \) is calculated as: \[ \sigma = \sqrt{npq} = \sqrt{1000 \cdot 0.25 \cdot 0.75} \approx 13.6931 \]

To find the probability that 220 or fewer families have 2 girls, we first convert the value to a Z-score. The Z-score for \( x = 220 \) is calculated as: \[ Z_{end} = \frac{x - \mu}{\sigma} = \frac{220 - 250}{13.6931} \approx -2.1909 \]

The probability \( P \) that 220 or fewer families will have 2 girls is given by: \[ P = \Phi(Z_{end}) - \Phi(-\infty) = \Phi(-2.1909) - 0 \] Using the standard normal distribution table, we find: \[ P \approx 0.0142 \]

Final Answer