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When dealing with probability distributions, it's essential to understand how different outcomes are associated with probabilities. For this exercise on dishwasher soap box sizes, we define a discrete probability distribution that relates the package sizes to their likelihood of selection.

• For a 25 oz box, the probability is 0.20.
• A 40 oz box has a probability of 0.50.
• The 65 oz box is selected with a probability of 0.30.

These probabilities must sum to 1, ensuring that every possible outcome has been considered. In this setting, we associate each box size with its respective probability to calculate key statistical measures of the distribution.
Understanding these probabilities lays the foundation for further analysis, such as expected values and variance.

The concept of sampling distribution focuses on the distribution of a statistic, such as the sample mean, calculated from all possible samples of the same size from a population. In this exercise, we examine the sampling distribution of the mean package size selected by two independent purchasers.
Each sample mean, denoted as \(\bar{X}\), is calculated by taking the average of two observed package sizes. Given the linearity of expectations, the expected value of the sample mean \(E(\bar{X})\) is equal to the expected value of a single selection, \(E(X)\). This makes it straightforward to determine the expected value for \(\bar{X}\), which turns out to be 44.5 oz.
The sampling distribution helps to understand and anticipate the behavior of sample means when drawing samples from a population. It's crucial in fields like statistics and data analysis to make generalizations from sample data alone.

Calculating variance is a key step in understanding the spread or variability in a dataset. For the probability distribution defined by the soap box sizes, variance gives us insight into how much the box sizes deviate from the average size.
To calculate the variance \(\sigma^2\) for the given distribution, use the formula:\[\sigma^2 = \sum (x_i - E(X))^2 p_i\]This involves computing the squared difference between each box size and the mean size, multiplied by the probability of that size. Summing these values gives a variance of 64.125, indicating the extent of variability in box size selections.
Understanding variance is vital as it provides a measure of dispersion, helping to make informed decisions based on data variability.

In statistics, an unbiased estimator is a statistic used to estimate a population parameter that, on average, equals the actual parameter value. For the dish soap sizes, we use the sample variance \(S^2\) as an unbiased estimator for the population variance \(\sigma^2\).
The formula for sample variance in a two-sample case simplifies due to the degrees of freedom being \(n-1\); hence, it's calculated as:\[S^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2\]Here, \(E(S^2)\) equals the population variance \(\sigma^2\), indicating that the sample variance is reliable as an unbiased estimator. Our calculations confirmed this as both \(E(S^2)\) and \(\sigma^2\) equal 64.125.
This quality of being unbiased is crucial in ensuring that sample statistics reliably represent their respective population parameters, facilitating accurate conclusions from statistical analyses.