In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.


Sampling Distribution of the Sample Variance


a. Find the value of the population variance σ².

Verified step by step guidance

1. Step 1: Recall the formula for population variance (σ²), which is given by: ${\sigma }^2=\frac{\sum (x-\mu {)}^2}{N},\; where\; \mu \; is\; the\; population\; mean,\; x\; represents\; each\; data\; point,\; and\; N\; is\; the\; population\; size.$
2. Step 2: Calculate the population mean (μ) using the formula: $\mu =\frac{\sum }{x}.\; For\; the\; population\; \{4,\; 5,\; 9\},\; sum\; the\; values\; (4\; +\; 5\; +\; 9)\; and\; divide\; by\; the\; population\; size\; (N\; =\; 3).$
3. Step 3: Subtract the population mean (μ) from each data point in the population to find the deviations: (x - μ). Then, square each deviation to get (x - μ)².
4. Step 4: Sum all the squared deviations obtained in Step 3. This gives the numerator of the variance formula: $\sum (x-\mu {)}^2$.
5. Step 5: Divide the sum of squared deviations by the population size (N = 3) to calculate the population variance (σ²).

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 data point and the mean. For the population {4, 5, 9}, the variance quantifies the spread of these values, providing insight into the variability within the entire population.

Sampling Distribution

The sampling distribution of a statistic, such as the sample variance, describes the distribution of that statistic across all possible samples of a given size from a population. When samples are taken with replacement, each sample can yield different values, and the sampling distribution helps to understand the variability and expected behavior of the sample variance as more samples are drawn.

Random Sampling with Replacement

Random sampling with replacement means that each time a sample is drawn from the population, the selected element is returned to the population before the next draw. This method ensures that each element has an equal chance of being selected in every draw, which is crucial for maintaining the independence of samples and for accurately estimating population parameters like variance.