In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.


Gender Selection Assume that the groups consist of 36 couples.


a. Find the mean and standard deviation for the numbers of girls in groups of 36 births.

Verified step by step guidance

1. Step 1: Recognize that this is a binomial probability distribution problem. The number of trials (n) is 36 (since there are 36 couples), and the probability of success (p) is 0.5 (since the probability of having a girl is 0.5).
2. Step 2: Recall the formula for the mean (μ) of a binomial distribution: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>μ</mi><mo>=</mo><mi>n</mi><mo>×</mo><mi>p</mi></math>. Substitute n = 36 and p = 0.5 into the formula to calculate the mean.
3. Step 3: Recall the formula for the standard deviation (σ) of a binomial distribution: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>σ</mi><mo>=</mo><msqrt><mi>n</mi><mo>×</mo><mi>p</mi><mo>×</mo><mo>(</mo><mn>1</mn><mo>-</mo><mi>p</mi><mo>)</mo></msqrt></math>. Substitute n = 36 and p = 0.5 into the formula to calculate the standard deviation.
4. Step 4: Simplify the expressions for both the mean and the standard deviation. For the mean, multiply n and p. For the standard deviation, calculate the product of n, p, and (1 - p), then take the square root of the result.
5. Step 5: Interpret the results. The mean represents the expected number of girls in 36 births, and the standard deviation measures the variability in the number of girls across different groups of 36 births.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, each birth can be seen as a trial where having a girl is a success, and the probability of having a girl is 0.5. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).

Mean of a Binomial Distribution

The mean of a binomial distribution is calculated using the formula μ = n * p, where n is the number of trials and p is the probability of success. For this problem, with 36 couples and a probability of 0.5 for having a girl, the mean number of girls expected can be easily computed. This value represents the average outcome over many trials.

Standard Deviation of a Binomial Distribution

The standard deviation of a binomial distribution measures the variability of the number of successes and is calculated using the formula σ = √(n * p * (1 - p)). This formula accounts for both the number of trials and the probabilities of success and failure. In this scenario, it helps to understand how much the actual number of girls born might deviate from the mean.