Questions: Let a population consist of the values 8 cigarettes, 21 cigarettes, and 22 cigarettes smoked in a day. 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? Calculate the mean absolute deviation for each possible sample of size 2 from the population. Sample Mean Absolute Deviation 8,8 0 8,21 6.5 8,22 7 21,8 6.5 21,21 0 21,22 0.5 22,8 7 22,21 0.5 22,22 0 (Type integers or decimals rounded to one decimal place as needed.) Calculate the mean of the sample mean absolute deviations. The mean of the sample mean absolute deviations is (Type an integer or decimal rounded to one decimal place as needed.)

Let a population consist of the values 8 cigarettes, 21 cigarettes, and 22 cigarettes smoked in a day. 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? Calculate the mean absolute deviation for each possible sample of size 2 from the population. Sample Mean Absolute Deviation 8,8 0 8,21 6.5 8,22 7 21,8 6.5 21,21 0 21,22 0.5 22,8 7 22,21 0.5 22,22 0 (Type integers or decimals rounded to one decimal place as needed.) Calculate the mean of the sample mean absolute deviations. The mean of the sample mean absolute deviations is (Type an integer or decimal rounded to one decimal place as needed.)
Transcript text: Let a population consist of the values 8 cigarettes, 21 cigarettes, and 22 cigarettes smoked in a day. 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? Calculate the mean absolute deviation for each possible sample of size 2 from the population. \begin{tabular}{|c|c|} \hline Sample & Mean Absolute Deviation \\ \hline$\{8,8\}$ & 0 \\ \hline$\{8,21\}$ & 6.5 \\ \hline$\{8,22\}$ & 7 \\ \hline$\{21,8\}$ & 6.5 \\ \hline$\{21,21\}$ & 0 \\ \hline$\{21,22\}$ & 0.5 \\ \hline$\{22,8\}$ & 7 \\ \hline$\{22,21\}$ & 0.5 \\ \hline$\{22,22\}$ & 0 \\ \hline \end{tabular} (Type integers or decimals rounded to one decimal place as needed.) Calculate the mean of the sample mean absolute deviations. The mean of the sample mean absolute deviations is $\square$ (Type an integer or decimal rounded to one decimal place as needed.)
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Solution

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Solution Steps

Step 1: Calculate Population MAD

To calculate the population MAD, first compute the mean (\(\mu\)) of the population values. Then, calculate the absolute deviations of each population value from the mean, and compute the mean of these absolute deviations. The population MAD is calculated as \(\frac{1}{N}\sum_{i=1}^{N}|v_i - \mu|\), where \(N\) is the population size. In this case, the population MAD is 6.

Step 2: Generate All Possible Samples

Since the sampling is with replacement, all possible pairs of population values are created, including pairs of the same value. This represents all possible samples of size 2.

Step 3: Calculate Sample MAD for Each Possible Sample

For each sample, calculate the mean. Then, compute the absolute deviation of each sample value from the sample mean, and find the mean of these absolute deviations to get the sample MAD.

Step 4: Calculate the Mean of Sample MADs

The mean of the MADs calculated for all possible samples is 3.1.

Step 5: Compare Sample MAD Mean with Population MAD

The mean of the sample MADs does not align with the population MAD, indicating that sample MAD may not be a reliable estimator for population MAD in all cases.

Final Answer:

The population MAD is 6, and the mean of the sample MADs is 3.1. This does not align with the population MAD, demonstrating the limitations of using sample MAD as an estimator for population MAD.

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