Mean Absolute Deviation Use the same population of {9 cigarett...

Question

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?

Accepted Answer

Welcome back, everyone. In this problem, consider a population with the following values 50, 80s, and 12 hours. Suppose samples of size 2 are randomly selected with replacement. After calculating the mean absolute deviations or MADs of the population and the samples, it is observed that the sample MADs do not center on the population MAD. What does this indicate about using the sample MAD as an estimator of the population MAD? A says the sample is always an unbiased estimator of the population MAD. B says the sample tends to be larger than the population, making it an overestimate. C says the sample does not reliably estimate the population because it is biased, and the D says the sample is always equal to the population for any random sample. Now what are we talking about when we say the mean absolute deviations are the MADs? Well, this measures the average distance of data points from the mean, and when calculating the MAD for a population, it considers all data points ensuring an accurate measure of variability. However, when we take random samples, the Computed sample MAD does not always accurately reflect the population MAD, which we can observe in this case because in this case we drew samples of size 2 with or sorry with replacement randomly and based on our problem we were told that the sample MADs do not center around the population MADs. So why would that happen? What's what's the reason behind that? Well, because of that, first it tells us there is bias in the sample MAD calculation. We know that small samples tend to have less variability than the full population. So since the mean absolute deviation depends on distances from the mean, using fewer data points reduces these distances on average. Another thing it tells us is that the sample MAD underestimates the population MAD because when computing MED for small samples, extreme values are less likely to be included, making the calculated MED smaller than the true population MAD in many cases. This means the sample MAD is a biased estimator of the population MAD, and that must mean C is the correct answer. The sample sample MAD does not reliably estimate the population MAD. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Mean Absolute Deviation (MAD)

Sampling Distribution

Estimator Bias

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