Answer :
To determine if the sample median is an unbiased estimator, we need to analyze the sample medians in comparison to the true population parameters.
Here are the steps to solve the problem:
1. Calculate the mean of the sample medians:
- The given sample medians from all possible samples of size 2 are: 18, 17.5, 17.5, 17, 17, 16.5, 16.5, 16.5, 16, and 15.
- Calculating the mean of these sample medians:
[tex]\[ \text{Mean of sample medians} = \frac{18 + 17.5 + 17.5 + 17 + 17 + 16.5 + 16.5 + 16.5 + 16 + 15}{10} = 16.75 \][/tex]
2. Calculate the median of the sample medians:
- Sorting the sample medians: 15, 16, 16.5, 16.5, 16.5, 17, 17, 17.5, 17.5, 18. The median is the middle value when ordered.
- Since there are 10 values, the median is the average of the 5th and 6th values:
[tex]\[ \text{Median of sample medians} = \frac{16.5 + 17}{2} = 16.75 \][/tex]
3. Compare the mean and median of sample medians to the population statistics:
- The mean age of the officers is:
[tex]\[ \text{Mean of officers' ages} = \frac{18 + 18 + 17 + 16 + 15}{5} = 16.8 \][/tex]
- The median age of the officers’ ages is 17.0 as given in the problem.
4. Determine if the sample median is an unbiased estimator:
- The mean of the sample medians is 16.75.
- The median of the sample medians is 16.75.
- The mean age of the officers is 16.8.
- The median age of the officers is 17.0.
Based on the criteria given in the problem, we analyze the statements as follows:
- Statement 1: "Yes, [tex]$50\%$[/tex] of the sample medians are 17 or more, and [tex]$50\%$[/tex] are below."
- This is a statement about the distribution, not about unbiased estimation properties.
- Statement 2: "Yes, the mean of the sample medians is 16.8, which is the same as the mean age of the officers."
- This is incorrect because the mean of the sample medians is 16.75, which is not the same as 16.8.
- Statement 3: "No, the mean of the sample medians is 16.8, which is not the same as the median age of the officers."
- This is incorrect because the mean of the sample medians is 16.75, not 16.8, and it doesn't compare properly to the median age.
- Statement 4: "No, the median of the sample medians is 16.75, which is not the same as the median age of the officers."
- This is accurate because it correctly states that the median of the sample medians (16.75) is not equal to the median age of the officers (17.0).
Thus, the correct answer is:
No, the median of the sample medians is 16.75, which is not the same as the median age of the officers.
Here are the steps to solve the problem:
1. Calculate the mean of the sample medians:
- The given sample medians from all possible samples of size 2 are: 18, 17.5, 17.5, 17, 17, 16.5, 16.5, 16.5, 16, and 15.
- Calculating the mean of these sample medians:
[tex]\[ \text{Mean of sample medians} = \frac{18 + 17.5 + 17.5 + 17 + 17 + 16.5 + 16.5 + 16.5 + 16 + 15}{10} = 16.75 \][/tex]
2. Calculate the median of the sample medians:
- Sorting the sample medians: 15, 16, 16.5, 16.5, 16.5, 17, 17, 17.5, 17.5, 18. The median is the middle value when ordered.
- Since there are 10 values, the median is the average of the 5th and 6th values:
[tex]\[ \text{Median of sample medians} = \frac{16.5 + 17}{2} = 16.75 \][/tex]
3. Compare the mean and median of sample medians to the population statistics:
- The mean age of the officers is:
[tex]\[ \text{Mean of officers' ages} = \frac{18 + 18 + 17 + 16 + 15}{5} = 16.8 \][/tex]
- The median age of the officers’ ages is 17.0 as given in the problem.
4. Determine if the sample median is an unbiased estimator:
- The mean of the sample medians is 16.75.
- The median of the sample medians is 16.75.
- The mean age of the officers is 16.8.
- The median age of the officers is 17.0.
Based on the criteria given in the problem, we analyze the statements as follows:
- Statement 1: "Yes, [tex]$50\%$[/tex] of the sample medians are 17 or more, and [tex]$50\%$[/tex] are below."
- This is a statement about the distribution, not about unbiased estimation properties.
- Statement 2: "Yes, the mean of the sample medians is 16.8, which is the same as the mean age of the officers."
- This is incorrect because the mean of the sample medians is 16.75, which is not the same as 16.8.
- Statement 3: "No, the mean of the sample medians is 16.8, which is not the same as the median age of the officers."
- This is incorrect because the mean of the sample medians is 16.75, not 16.8, and it doesn't compare properly to the median age.
- Statement 4: "No, the median of the sample medians is 16.75, which is not the same as the median age of the officers."
- This is accurate because it correctly states that the median of the sample medians (16.75) is not equal to the median age of the officers (17.0).
Thus, the correct answer is:
No, the median of the sample medians is 16.75, which is not the same as the median age of the officers.
