Mean Absolute Deviation Use the same population of {9 cigarettes, 10 cigarettes, 20 cigarettes} from Exercise 45. Show that when samples of size 2 are randomly selected with replacement, the samples have mean absolute deviations that do not center about the value of the mean absolute deviation of the population. What does this indicate about a sample mean absolute deviation being used as an estimator of the mean absolute deviation of a population?

Verified step by step guidance

1. Step 1: Calculate the mean absolute deviation (MAD) of the population. First, find the mean of the population by summing all the values and dividing by the number of values. Then, calculate the absolute deviation of each value from the mean, and finally, find the average of these absolute deviations.
2. Step 2: List all possible samples of size 2 that can be selected with replacement from the population {9, 10, 20}. For example, the samples include (9, 9), (9, 10), (9, 20), (10, 9), (10, 10), (10, 20), (20, 9), (20, 10), and (20, 20).
3. Step 3: For each sample, calculate the sample mean. Then, compute the absolute deviation of each value in the sample from the sample mean. Finally, calculate the mean absolute deviation (MAD) for each sample by averaging these absolute deviations.
4. Step 4: Compare the mean absolute deviations of the samples to the population MAD. Observe whether the sample MADs center around the population MAD. This involves analyzing the distribution of the sample MADs and comparing it to the population MAD.
5. Step 5: Conclude that the sample MADs do not center around the population MAD. This indicates that the sample MAD is a biased estimator of the population MAD, meaning it does not reliably estimate the population MAD when used in this context.

Key Concepts

Mean Absolute Deviation (MAD)

Mean Absolute Deviation is a measure of dispersion that quantifies the average distance between each data point in a dataset and the mean of that dataset. It is calculated by taking the absolute differences between each data point and the mean, summing these differences, and then dividing by the number of data points. MAD provides insight into the variability of the data, making it useful for understanding how spread out the values are around the mean.

Sampling Distribution

The sampling distribution refers to the probability distribution of a statistic (like the sample mean or sample MAD) obtained from a large number of samples drawn from a specific population. When samples are taken with replacement, the sampling distribution can differ from the population distribution, leading to variations in the calculated statistics. Understanding this concept is crucial for interpreting how sample statistics can estimate population parameters.

Estimator Bias

Estimator bias occurs when a statistical estimator consistently overestimates or underestimates a population parameter. In the context of mean absolute deviation, if the sample MAD does not center around the population MAD, it indicates that the sample may not be a reliable estimator of the population's true variability. Recognizing bias is essential for evaluating the accuracy and reliability of statistical estimates in inferential statistics.