Geometric Distribution: Mean and Variance In Exercises 29 and 30, use the fact that the mean of a geometric distribution is μ = 1/p and the variance is
sigma^2 = q/p^2
Paycheck Errors A company assumes that 0.5% of the paychecks for a year were calculated incorrectly. The company has 200 employees and examines the payroll records from one month. (a) Find the mean, variance, and standard deviation. (b) How many employee payroll records would you expect to examine before finding one with an error?

Verified step by step guidance

1. Step 1: Understand the problem. The problem involves a geometric distribution, where the probability of success (finding an error) is p = 0.005 (0.5%). The formulas for the mean, variance, and standard deviation of a geometric distribution are given as follows: Mean (μ) = 1/p, Variance (σ²) = q/p², and Standard Deviation (σ) = √(Variance). Here, q = 1 - p is the probability of failure.
2. Step 2: Calculate the mean (μ). Using the formula μ = 1/p, substitute p = 0.005 into the equation. This will give you the expected number of payroll records to examine before finding one with an error.
3. Step 3: Calculate the variance (σ²). Using the formula σ² = q/p², first calculate q = 1 - p = 1 - 0.005 = 0.995. Then substitute q and p into the variance formula to compute the variance.
4. Step 4: Calculate the standard deviation (σ). Using the formula σ = √(Variance), take the square root of the variance calculated in the previous step to find the standard deviation.
5. Step 5: Interpret the results. The mean represents the expected number of payroll records to examine before finding one with an error. The variance and standard deviation provide measures of the spread or variability in the number of records examined before finding an error.

Key Concepts

Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. It is characterized by a constant probability of success, denoted as 'p'. The mean of this distribution is calculated as μ = 1/p, indicating the average number of trials expected before the first success occurs.

Mean and Variance

In statistics, the mean represents the average value of a dataset, while variance measures the spread or dispersion of the data points around the mean. For a geometric distribution, the mean is μ = 1/p and the variance is σ² = q/p², where q = 1 - p. Understanding these concepts is crucial for analyzing the expected outcomes and variability in scenarios modeled by the geometric distribution.

Standard Deviation

Standard deviation is the square root of the variance and provides a measure of the average distance of each data point from the mean. It is a key indicator of variability in a dataset. In the context of the geometric distribution, calculating the standard deviation helps to understand the typical deviation from the mean number of trials needed to find the first success, which is essential for interpreting the results of the payroll error analysis.