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.)


c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by n); then find the mean of those nine population variances.

Verified step by step guidance

1. Step 1: Identify the population values and calculate the population mean. The population consists of the values 9, 10, and 20 cigarettes. The population mean (μ) is calculated as the sum of all population values divided by the number of values in the population. Use the formula: μ = (Σx) / N, where Σx is the sum of the population values and N is the number of values in the population.
2. Step 2: List all possible samples of size 2 with replacement. Since the population has 3 values (9, 10, 20), and sampling is done with replacement, there are 3 × 3 = 9 possible samples. These samples are: (9, 9), (9, 10), (9, 20), (10, 9), (10, 10), (10, 20), (20, 9), (20, 10), (20, 20).
3. Step 3: For each sample, calculate the sample mean. For a sample (x₁, x₂), the sample mean (x̄) is calculated as: x̄ = (x₁ + x₂) / 2. Perform this calculation for all 9 samples.
4. Step 4: For each sample, calculate the population variance using the formula for population variance: σ² = (Σ(xᵢ - x̄)²) / n, where xᵢ are the sample values, x̄ is the sample mean, and n is the sample size (n = 2 in this case). Perform this calculation for all 9 samples.
5. Step 5: Find the mean of the 9 population variances. Add up all 9 variances calculated in Step 4 and divide by 9 to find the mean variance. This mean variance represents the average of the variances when treating each sample as a population.

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 using the formula: σ² = Σ(xi - μ)² / N, where xi represents each value, μ is the population mean, and N is the total number of values. This concept is crucial for understanding how to quantify the spread of data points in a population.

Sampling with Replacement

Sampling with replacement means that after a value is selected from a population, it is returned to the population before the next selection. This method allows for the same value to be chosen multiple times, which affects the calculation of statistics like variance and mean. Understanding this concept is essential for correctly analyzing the samples drawn from the population.

Mean of Variances

The mean of variances involves calculating the average of multiple variance values obtained from different samples. In this context, after calculating the population variance for each of the nine possible samples, the mean of these variances provides insight into the overall variability of the samples. This concept helps in understanding the consistency of variance across different sample selections.