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Chapter 9: Problem 13

If two statistics are available for estimating a population characteristic, under what circumstances might you choose a biased statistic over an unbiased statistic?

Short Answer

Expert verified
You might choose a biased statistic over an unbiased statistic for estimating a population characteristic under circumstances such as: 1. Smaller variance: The biased statistic provides more precise and stable estimates with less uncertainty, even if they are systematically off from the true value. 2. Reduced computational complexity: The biased statistic is simpler to compute, making it a more practical choice when computational resources are limited. 3. Specific purpose: The analysis requires estimation of a specific population characteristic rather than an overall average, making a biased statistic more suitable. 4. Limited data availability: Biased statistics may be more appropriate when there is limited data since they can use available data more efficiently than unbiased statistics.

Step by step solution

01

Define Biased and Unbiased Statistics

A statistic is considered unbiased if, on average, it provides the true value of the population parameter it is estimating. In other words, an unbiased statistic's expected value is equal to the true value of the population parameter. On the other hand, a biased statistic is a statistic whose expected value is not equal to the true value of the population parameter, leading to a consistent over- or underestimation of the parameter.
02

The Differences Between Biased and Unbiased Statistics

Biased statistics lead to systematic errors in estimating the true population parameter, causing the estimates to consistently deviate from the actual value. Unbiased statistics, on the other hand, provide estimates that are, on average, equal to the true value of the population parameter. However, unbiased statistics might still have larger variances or more uncertainty compared to biased statistics. Therefore, the choice between biased and unbiased statistics depends on the context and the priorities of the analysis.
03

Circumstances for Choosing a Biased Statistic over an Unbiased Statistic

Some situations where one might prefer a biased statistic over an unbiased statistic for estimating a population characteristic are: 1. **Smaller Variance**: If the biased statistic has a significantly smaller variance than the unbiased statistic, it can provide more precise and stable estimates, even if they are systematically off from the true value. In such cases, the trade-off between bias and variance might lead to a preference for a biased statistic. 2. **Reduced Computational Complexity**: In some situations, the unbiased statistic might be computationally intensive or difficult to calculate. A biased statistic, on the other hand, might be relatively simple to compute, making it a more practical choice in scenarios where computational resources are limited. 3. **Specific Purpose**: When the purpose of the analysis requires the estimation of a specific population characteristic rather than an overall average (e.g. representing a subpopulation, or predicting some extreme cases), a biased statistic might be more suitable. 4. **Limited Data Availability**: In situations where there is limited data availability, biased statistics might be more appropriate, as they can make use of the available data more efficiently than unbiased statistics. In conclusion, although unbiased statistics are generally preferred for estimating population characteristics, there are certain circumstances where a biased statistic might be a more appropriate choice. In these cases, the trade-off between bias and variance, computational complexity, specific purposes, and data availability should be considered when deciding which statistic to use.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Bias
Understanding statistical bias is fundamental for interpreting the results of any analysis. Statistical bias occurs when an estimator鈥攁 statistic used to infer the value of a population parameter鈥攄oes not equal the true population value on average. This can happen when the data collection method is flawed or when there is a problem with the estimator itself. Biased estimators consistently over- or under-estimate the parameter, leading to systematic errors in your conclusions. For example, a survey that consistently under-reports a certain group's opinions would yield a biased estimate of the overall population's sentiment.

Biased statistics can still be valuable in certain contexts, especially when unbiased estimators are not feasible to calculate or might come with high variability.
Expected Value of Statistic
The expected value of a statistic signifies the mean of the distribution of that statistic. When dealing with estimators, the expected value is crucial because it determines the statistical bias of the estimator. An estimator is considered unbiased if its expected value matches the true population parameter it aims to estimate. Otherwise, it is biased.

For a practical scenario, suppose we have an estimator that predicts the average height of a population. If the estimated average height consistently matches the real average height over multiple samples, the estimator has an expected value equal to the true average height, making it unbiased.
Variance of Estimator
Variance is a measure of how much the estimates produced by a statistic will vary from one sample to another. Low variance means the estimates tend to be very close to each other; high variance indicates they are spread out. For estimators, which are statistics used to infer the value of a population parameter, a high variance can diminish the reliability of the estimate even if the estimator is unbiased.

Considering an unbiased estimator with high variance, despite being correct on average, it can widely miss the mark for individual estimates. For instance, if we measure the intelligence of a group of animals with an unbiased estimator, high variance may lead to some estimates suggesting significantly higher or lower intelligence than what is accurate for individual animals, thus affecting the precision of the conclusions.
Computational Complexity in Statistics
In statistical analysis, computational complexity refers to the amount of computational power and time required to calculate a statistic or estimator. In the real world, resources are often limited, making the computational complexity an important consideration. Unbiased estimators can sometimes be computationally expensive and may not be feasible for large datasets or when quick computation is necessary.

For instance, computing an unbiased estimator for a complex model may involve intricate calculations that strain computational resources, whereas a simpler biased estimator could offer substantial savings in time and computing power, making it more suitable in certain practical applications.
Trade-off Between Bias and Variance
The trade-off between bias and variance is a central theme in the selection of statistical methods. An unbiased estimator does not guarantee it's the best choice, as it might come with high variance, resulting in less reliable individual predictions. Conversely, a biased estimator with low variance can often give more consistent results.

In practice, you need to strike a balance depending on the context of the study. If having stable, repeatable results is more critical, you may prefer to accept some bias in exchange for lower variance. This trade-off is vital when data is limited, or when the cost (in terms of time or computation) of reducing bias is high compared to the benefit of reducing variance.

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Most popular questions from this chapter

The paper "Sleeping with Technology: Cognitive, Affective and Technology Usage Predictors of Sleep Problems Among College Students" (Sleep Health [2016]: 49-56) summarized data from a survey of a sample of college students. Of the 734 students surveyed, 125 reported that they sleep with their cell phones near the bed and check their phones for something other than the time at least twice during the night. For purposes of this exercise, assume that this sample is representative of college students in the United States. a. Use the given information to estimate the proportion of college students who check their cell phones for something other than the time at least twice during the night. b. Verify that the conditions needed in order for the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in the context of this problem.

The report "The 2016 Consumer Financial Literacy Survey" (The National Foundation for Credit Counseling, www .nfcc.org, retrieved October 28,2016 ) summarized data from a representative sample of 1668 adult Americans. When asked if they typically carry credit card debt from month to month, 584 of these people responded "yes." a. Use the given information to estimate the proportion of adult Americans who carry credit card debt from month to month. b. Verify that the conditions needed in order for the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in the context of this problem.

Appropriate use of the interval $$ \hat{p} \pm(z \text { critical value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ requires a large sample. For each of the following combinations of \(n\) and \(\hat{p}\), indicate whether the sample size is large enough for this interval to be appropriate. a. \(n=100\) and \(\hat{p}=0.70\) b. \(n=40\) and \(\hat{p}=0.25\) c. \(n=60\) and \(\hat{p}=0.25\) d. \(n=80\) and \(\hat{p}=0.10\)

For each of the following choices, explain which one would result in a wider large-sample confidence interval for \(p:\) a. \(90 \%\) confidence level or \(95 \%\) confidence level b. \(n=100\) or \(n=400\)

The formula used to calculate a large-sample confidence interval for \(p\) is $$ \hat{p} \pm(z \text { critical value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ What is the appropriate \(z\) critical value for each of the following confidence levels? a. \(90 \%\) b. \(99 \%\) c. \(80 \%\)
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