Finding and Interpreting Mean, Variance, and Standard Deviation In Exercises 31–36, find the mean, variance, and standard deviation of the binomial distribution for the given random variable. Interpret the results and determine any unusual values.


Late for Work Thirty-one percent of U.S. employees who are late for work blame oversleeping. You randomly select 12 U.S. employees who are late for work and ask them whether they blame oversleeping. The random variable represents the number who are late for work and blame oversleeping. (Source: CareerBuilder)

Verified step by step guidance

1. Step 1: Identify the parameters of the binomial distribution. The problem states that the probability of success (blaming oversleeping) is p = 0.31, the number of trials (employees selected) is n = 12, and the random variable X represents the number of employees who blame oversleeping.
2. Step 2: Calculate the mean (expected value) of the binomial distribution using the formula: $\mu =n\times p$. Substitute n = 12 and p = 0.31 into the formula.
3. Step 3: Calculate the variance of the binomial distribution using the formula: $\sigma {²}^{}=n\times p\times \; (1\; -p)$. Substitute n = 12 and p = 0.31 into the formula.
4. Step 4: Calculate the standard deviation by taking the square root of the variance: $\sigma =\; \surd (\sigma {²}^{})$. Use the variance calculated in Step 3.
5. Step 5: Interpret the results. The mean represents the average number of employees who blame oversleeping out of 12. The standard deviation measures the typical deviation from the mean. To determine unusual values, use the rule of thumb that values more than 2 standard deviations away from the mean (either above or below) are considered unusual.

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, the random variable represents the number of employees who blame oversleeping among a sample of 12. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).

Mean, Variance, and Standard Deviation

The mean of a binomial distribution is calculated as n * p, representing the expected number of successes. Variance measures the spread of the distribution and is calculated as n * p * (1 - p). The standard deviation, the square root of the variance, provides insight into the typical distance of values from the mean, helping to understand the distribution's variability.

Interpreting Results

Interpreting the mean, variance, and standard deviation involves understanding what these statistics reveal about the data. The mean indicates the average number of employees blaming oversleeping, while the variance and standard deviation inform us about the consistency of this behavior. Identifying unusual values can help in assessing whether certain outcomes deviate significantly from what is expected based on the calculated statistics.