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Chapter 6: Problem 3

Data Set 4 "Births" in Appendix B includes birth weights of 400 babies. If we compute the values of sample statistics from that sample, which of the following statistics are unbiased estimators of the corresponding population parameters: sample mean; sample median; sample range; sample variance; sample standard deviation; sample proportion?

Short Answer

Expert verified
Unbiased estimators: sample mean, sample variance, sample proportion.

Step by step solution

01

Understand Unbiased Estimators

An unbiased estimator is a statistic that on average equals the parameter it estimates. In other words, the expected value of the estimator equals the population parameter.
02

Identify Each Statistic

List the given sample statistics: mean, median, range, variance, standard deviation, and proportion.
03

Sample Mean

The sample mean is an unbiased estimator of the population mean. Its expected value is equal to the population mean.
04

Sample Median

The sample median is not generally an unbiased estimator of the population median, especially for small sample sizes. However, for large samples, it can be approximately unbiased.
05

Sample Range

The sample range is a biased estimator of the population range. The sample range systematically underestimates the population range, especially for small sample sizes.
06

Sample Variance

The sample variance, calculated as \frac{1}{n-1}\times \(\text{sum of squared deviations from the mean}\) (with Bessel's correction), is an unbiased estimator of the population variance.
07

Sample Standard Deviation

The sample standard deviation is a biased estimator of the population standard deviation. It underestimates the population value.
08

Sample Proportion

The sample proportion is an unbiased estimator of the population proportion. Its expected value equals the population proportion.

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Key Concepts

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

Sample Mean
The sample mean, often denoted as \(\bar{x}\), is calculated by adding up all the values in a sample and dividing by the number of values. It serves as an unbiased estimator of the population mean \(\mu\). This means that the expected value of the sample mean equals the population mean. In simpler terms, if you took many samples and calculated their means, those means would average out to the true population mean. This makes \(\bar{x}\) a reliable measure when trying to estimate \(\mu\). For example, if you calculated the mean birth weight of 400 babies, this would give you an accurate estimate of the average birth weight for the entire population.
Sample Variance
Sample variance, denoted as \(s^2\), measures the spread or variability of data points in a sample. It is calculated using the formula: \[ s^2 = \frac{1}{n-1} \times \text{sum of squared deviations from the mean} \] Here, \(n-1\) is used instead of \(n\) to account for the degrees of freedom, known as Bessel's correction. This correction makes the sample variance an unbiased estimator of the population variance \(\sigma^2\). Without this correction, the variance would underestimate the true variability in the population.
Sample Proportion
The sample proportion is used to estimate the proportion of a characteristic in a population. For example, if you want to know the proportion of babies born with a certain birth weight, you would use the sample proportion. Mathematically, it is represented as \(\hat{p}\), and is calculated by dividing the number of successes (or characteristics of interest) by the total sample size \(n\). \(\hat{p}\) is considered an unbiased estimator of the population proportion \(p\) because its expected value is equal to \(p\). This makes \(\hat{p}\) a useful statistic when dealing with proportions.
Statistical Bias
Statistical bias occurs when an estimator does not equal the population parameter on average. In other words, an estimator is biased if its expected value is not equal to the parameter it estimates. For example:
  • The sample range is a biased estimator of the population range because it tends to underestimate the true range, especially in small samples.
  • The sample standard deviation is also a biased estimator of the population standard deviation since it typically underestimates the actual variability of the population.
Understanding bias is crucial for making accurate inferences about a population from a sample.
Population Parameters
Population parameters are numerical characteristics of a population, such as the mean \(\mu\), variance \(\sigma^2\), and proportion \(p\). These parameters summarize information about an entire population. In practice, it is often difficult or impossible to measure every member of a population. Therefore, we use sample statistics to estimate these parameters. The goal is to use unbiased estimators, ensuring that our sample statistics provide accurate estimates of the population parameters. For instance:
  • The sample mean \(\bar{x}\) estimates the population mean \(\mu\)
  • The sample variance \(s^2\) estimates the population variance \(\sigma^2\)
  • The sample proportion \(\hat{p}\) estimates the population proportion \(p\)
By understanding and using unbiased estimators, we can make more reliable conclusions about the population based on sample data.

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Most popular questions from this chapter

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2. About ________ \(\%\) of the area is between \(z=-2\) and \(z=2\) (or within 2 standard deviations of the mean).

Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of \(1 .\) In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places. Find \(P_{99}\), the 99 th percentile. This is the bone density score separating the bottom \(99 \%\) from the top \(1 \%\).

Use the data in the table below for sitting adult males and females (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. (Hint: Draw a graph in each case.) $$ \begin{aligned} &\text { Sitting Back-to-Knee Length (inches) }\\\ &\begin{array}{l|c|c|c} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \text { in. } & 1.1 \text { in. } & \text { Normal } \\ \hline \text { Females } & 22.7 \text { in. } & 1.0 \text { in. } & \text { Normal } \\ \hline \end{array} \end{aligned} $$ Instead of using \(0.05\) for identifying significant values, use the criteria th?t a value \(x\) is significantly high if \(P(x\) or greater \() \leq 0.01\) and a value is significantly low if \(P(x\) or less \() \leq 0.01\). Find the back-to- knee lengths for males, separating significant values from those that are not significant. Using these criteria, is a male back-to-knee length of 26 in. significantly high?

The University of Maryland Medical Center considers "low birth weights" to be those that are less than \(5.5 \mathrm{lb}\) or \(2495 \mathrm{~g} .\) Birth weights are normally distributed with a mean of \(3152.0 \mathrm{~g}\) and a standard deviation of \(693.4 \mathrm{~g}\) (based on Data Set 4 "Births" in Appendix B). a. If a birth weight is randomly selected, what is the probability that it is a "low birth weight"? b. Find the weights considered to be significantly low, using the criterion of a birth weight having a probability of \(0.05\) or less. c. Compare the results from parts (a) and (b).

Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2. About __________\(\%\) of the area is between \(z=-1\) and \(z=1\) (or within 1 standard deviation of the mean).
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