3. Describing Data Numerically
Mean
1:50 minutesProblem 6.1.2
Textbook Question
Which statistic is the best unbiased estimator for μ?
a. s
b. xbar
c. the median
d. the mode
Verified step by step guidance1
Understand the concept of an unbiased estimator: An unbiased estimator is a statistic whose expected value is equal to the parameter it is estimating. In this case, we are looking for the best unbiased estimator for the population mean (μ).
Review the options provided: (a) s (sample standard deviation), (b) x̄ (sample mean), (c) the median, and (d) the mode. Consider which of these statistics is most directly related to estimating the population mean.
Recall that the sample mean (x̄) is an unbiased estimator of the population mean (μ). This means that the expected value of x̄ is equal to μ, making it the best choice among the options.
Understand why the other options are not the best unbiased estimators: (a) The sample standard deviation (s) estimates the population standard deviation, not the mean. (c) The median and (d) the mode are measures of central tendency but are not guaranteed to be unbiased estimators of the population mean.
Conclude that the sample mean (x̄) is the best unbiased estimator for μ because it satisfies the condition of unbiasedness and is specifically designed to estimate the population mean.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unbiased Estimator
An unbiased estimator is a statistical estimator that, on average, equals the parameter it estimates. This means that if you were to take many samples and calculate the estimator for each sample, the average of those estimates would converge to the true parameter value. In this context, we are looking for an estimator of the population mean (μ) that does not systematically overestimate or underestimate it.
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Sample Mean (x̄)
The sample mean, denoted as x̄, is the average of a set of sample observations. It is calculated by summing all the sample values and dividing by the number of observations. The sample mean is a commonly used estimator for the population mean (μ) and is considered an unbiased estimator, making it a strong candidate in the context of the question.
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Other Measures of Central Tendency
Measures of central tendency, such as the median and mode, summarize a set of data points. The median is the middle value when data is ordered, while the mode is the most frequently occurring value. Although these measures provide insights into the data, they are not unbiased estimators of the population mean (μ) like the sample mean (x̄), which is why they are less suitable in this context.
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