Question

Consider a hypothetical population of dogs in which there are four possible weights, all of which are equally likely: \(42,48,52,\) or 58 pounds. If a sample of size \(n=2\) is drawn from this population, what is the sampling distribution of the total weight of the two dogs selected? That is, what are the possible values for the total and what are the probabilities associated with each of those values?

Short answer

The possible total weights and their associated probabilities for a sample of two dogs are: 84 lbs (p = 1/16), 90 lbs (p = 1/8), 96 lbs (p = 3/16), 100 lbs (p = 1/8), 104 lbs (p = 1/4), 110 lbs (p = 1/8), 116 lbs (p = 3/16), 120 lbs (p = 1/8), and 126 lbs (p = 1/16).

Step-by-step solution

Identify Possible Weights

The first step is to identify the possible weights of the dogs in the population, which are already given as 42, 48, 52, and 58 pounds. Each of these weights is equally likely, meaning the probability of selecting a dog of any given weight is the same.

List All Possible Combinations

Since a sample size of two dogs is being selected, calculate the total weights possible by combining every two weights, including combining weights with themselves, since a weight could be picked more than once. The combinations are (42,42), (42,48), (42,52), (42,58), (48,42), (48,48), (48,52), (48,58), (52,42), (52,48), (52,52), (52,58), (58,42), (58,48), (58,52), and (58,58).

Calculate Total Weights

Calculate the sum of the two weights for each of the listed combinations to determine the possible total weights. For instance, for the combination of weights (42, 42), the total would be 42 + 42 = 84 pounds. Perform this operation for each pair of combinations.

Determine the Probabilities

Since all individual weights are equally likely, the probability of drawing any single weight is 1/4. The probability of drawing any combination of two weights is the product of their individual probabilities. For combinations of different weights (e.g., 42 and 48), the probability is (1/4) * (1/4) = 1/16. For combinations with the same weight (e.g., 42 and 42), the probability is also 1/16 because it's the same event occurring twice.

Combine and Simplify Probabilities

Some total weights can be achieved with different combinations; for each total weight, combine the probabilities of the individual combinations that give that total. For instance, the total weight of 84 pounds can be achieved only by the combination (42, 42), so its probability is 1/16, but a total weight of 90 pounds can be achieved by two different combinations: (42, 48) and (48, 42), thus the probability for a total of 90 would be 1/16 + 1/16 = 1/8.

Create the Sampling Distribution Table

Construct a table listing all the unique total weights and their corresponding probabilities. This table is the sampling distribution of the total weight of the two dogs selected.

Key concepts

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random events and quantifying the likelihood of occurrences. It's fundamental to statistical sampling and helps us understand and predict outcomes when dealing with uncertain events.

In the context of our dog weight example, probability theory is applied to evaluate the chance of drawing any specific combination of dog weights when a sample is selected. Since each weight is equally likely, we determine these probabilities using fundamental probability principles. For different weight combinations, the probability is the product of each individual weight probability because these are independent events; getting a certain weight on the first draw doesn't affect what we get on the second draw. To find the probability of selecting the same weight twice, we treat it as if the same single event is happening twice, which gives us the probability of one weight occurring multiplied by itself.

Ultimately, to calculate the probability of any outcome in our sampling distribution, we use the rule of multiplication for independent events and combine individual event probabilities when multiple combinations lead to the same total weight.

Combinatorics

Combinatorics is the area of mathematics that deals with counting combinations and arrangements of objects. It's instrumental in determining the number of possible outcomes in a statistical sample, especially when dealing with permutations and combinations.

Using combinatorics, we listed all potential combinations of two dog weights, considering that the same weight can be selected more than once. This comprehensive list of pairs is essential because each represents a possible outcome with its own chance of occurring. This approach is particularly useful in probability theory and sampling distribution exercises because it ensures that all possible scenarios are considered in our calculations of the total weights.

Understanding the basics of combinatorics helps to efficiently solve the exercise by systematically accounting for every possible way to draw a sample of two dogs from the given population. This combinatorial foundation is a stepping stone to accurately determining the sampling distribution in the next steps.

Statistical Sampling

Statistical sampling involves selecting a subset (a sample) from a larger group (a population) in order to estimate characteristics of the whole group. The sampling distribution is a probability distribution that arises when a statistic is computed from a sample of a certain size randomly drawn from a population.

In our exercise, we demonstrated statistical sampling by selecting two dogs from the population at random. We did this with an understanding that the total weight of these dogs was our statistic of interest. By computing the sum of the dog weights for all possible sampled pairs and assigning probabilities to these sums, we formed the sampling distribution of the total weight.

The sampling distribution gives us vital information, such as the mean and variance of our statistic, and in more complex scenarios, it can be used to make inferences about the population from which the sample was drawn. For example, it could help us predict the probability of the total weight of two randomly selected dogs exceeding a certain threshold, which is a common type of problem in statistical analysis.