Why Divide by ? Let a population consist of the values 9 cigarettes, 10 cigarettes, and 20 cigarettes smoked in a day (based on data from the California Health Interview Survey). Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.)


a. Find the variance of the population {9 cigarettes, 10 cigarettes, 20 cigarettes}.

Verified step by step guidance

1. Step 1: Understand the concept of variance. Variance measures the spread of data points in a population or sample. For a population, the variance formula is: \( \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \), where \( x_i \) are the individual data points, \( \mu \) is the population mean, and \( N \) is the number of data points in the population.
2. Step 2: Calculate the population mean \( \mu \). The mean is given by \( \mu = \frac{\sum x_i}{N} \), where \( x_i \) are the data points and \( N \) is the total number of data points. For the population \{9, 10, 20\}, sum the values and divide by the total number of values.
3. Step 3: Compute the squared deviations from the mean for each data point. For each \( x_i \), calculate \( (x_i - \mu)^2 \). This step involves subtracting the mean from each data point and squaring the result.
4. Step 4: Sum the squared deviations. Add up all the squared deviations calculated in the previous step. This gives the numerator of the variance formula.
5. Step 5: Divide the sum of squared deviations by \( N \), the total number of data points in the population. This final step yields the population variance \( \sigma^2 \).

Key Concepts

Population Variance

Population variance is a measure of how much the values in a population differ from the population mean. It is calculated by taking the average of the squared differences between each value and the mean. For a population with values {x1, x2, ..., xn}, the formula is σ² = Σ(xi - μ)² / N, where μ is the population mean and N is the number of values in the population.

Mean Calculation

The mean, or average, is a central value of a set of numbers, calculated by summing all the values and dividing by the count of those values. In the context of the given population, the mean is essential for determining how far each value deviates from the average, which is a critical step in calculating variance.

Sampling with Replacement

Sampling with replacement means that after a value is selected from the population, it is returned to the population before the next selection. This method ensures that each selection is independent and maintains the same probability distribution for each draw, which is important for understanding the behavior of sample statistics derived from the population.