Question

Mean and Standard Deviation of Boys or Girls In Exercise P.116, we discuss the random variable counting the number of girls in three babies, given that the proportion of babies who are girls is about \(0.49 .\) Find the mean and standard deviation of this random variable.

Short answer

The mean number of girls among three babies is 1.47, and the standard deviation is 0.8643.

Step-by-step solution

Calculation of the Mean

The mean (\(μ\)) of a binomial distribution is calculated by the formula \(μ = np\). For this case, \(n = 3\) (number of babies) and \(p = 0.49\) (probability of a baby being a girl). \(μ = np = 3 * 0.49 = 1.47\). So the mean number of girls is 1.47.

Calculation of the Variance

The variance (\(σ^2\)) of a binomial distribution is calculated by the formula \(σ^2 = np(1-p)\). For this case, \(n = 3\), \(p = 0.49\) and \(1 - p = 0.51\). \(σ^2 = np(1-p) = 3 * 0.49 * 0.51 = 0.747\).

Calculation of the Standard Deviation

The standard deviation (\(σ\)) is the square root of the variance. So, \(σ = \sqrt{σ^2} = \sqrt{0.747} = 0.8643\).

Key concepts

Mean of a Binomial Distribution

Understanding the mean, or expected value, of a binomial distribution is crucial for grasping the behavior of a binomially distributed random variable. In essence, the mean tells us what the average outcome would be if we were to repeat a binomial experiment a large number of times.

The mean of a binomial distribution is computed using the formula \(\mu = np\), where \(n\) represents the number of trials and \(p\) the probability of success on any given trial. For example, if we're looking at the number of girls in three babies, and we know the probability of a baby being a girl is approximately 0.49, we use these values in our formula. With \(n = 3\) and \(p = 0.49\), the calculation is \(\mu = 3 \times 0.49 = 1.47\). This result tells us that, on average, we expect about 1.47 girls in three babies.

Remember, the mean is just the 'center' of the distribution and doesn't give information about the variability of the outcomes. That's where the concepts of variance and standard deviation come into play.

Variance of a Binomial Distribution

While the mean provides a central tendency, the variance of a binomial distribution offers insight into its variability. Variance measures how much the outcomes of a random variable differ from the mean. A higher variance suggests that the results are more spread out.

The variance formula for a binomial distribution is \(\sigma^2 = np(1-p)\), where \(n\) again is the number of trials, \(p\) is the probability of a single success, and \(1-p\) is the probability of a single failure. Taking the same scenario with three babies, the calculation of the variance would be \(\sigma^2 = 3 \times 0.49 \times 0.51 = 0.747\). This figure indicates that there is a degree of spread in the number of girls we might expect when looking at groups of three babies.

The variance is a fundamental component because it not only informs us about the spread but also sets the stage for calculating the standard deviation, which is a more intuitive measure of variability for many people.

Standard Deviation of a Binomial Distribution

The standard deviation brings the concept of variance 'down to earth,' providing a clearer picture of variability. In a binomial distribution, while the variance shows the dispersion, the standard deviation does so in the same units as the data, which makes it more interpretable.

To calculate the standard deviation, we take the positive square root of the variance. Following our example, the standard deviation is obtained from the variance \(\sigma^2 = 0.747\) by computing \(\sigma = \sqrt{0.747} = 0.8643\). The standard deviation reveals that, the outcomes (number of girls in this case) tend to deviate from the mean (1.47 girls) by about 0.8643 girls when we look at many sets of three babies.

This measure is particularly useful because it helps to understand the range within which we can expect most outcomes to fall, assuming a normal distribution. And, because it's in the same units as our data - number of girls - it's easier to wrap our heads around than variance, which is in squared units.