According to the Fisher-Neyman factorization theorem, a statistic is sufficient for a parameter if the conditional distribution of the data given the statistic does not depend on the parameter. This implies that the statistic captures all the information about the parameter contained in the sample.
22
If the expectation of a function of a statistic is independent of the parameter, what is that statistic called?
A statistic T is defined as complete if the condition E[g(T)] = 0 for all parameters implies that g(T) = 0 with probability 1. This property is essential in the Lehmann-Scheffe theorem for finding the Uniformly Minimum Variance Unbiased Estimator (UMVUE).
23
A set of joint sufficient statistics is considered minimal if it functions as a component of any other sufficient statistic. What is the correct term for this?
A minimal sufficient statistic is a sufficient statistic that is a function of any other sufficient statistic. It provides the most compact summary of the data that still retains all the information necessary for estimating the parameter of interest. It is the 'smallest' possible sufficient statistic.
24
If μ represents the population parameter and x̅ represents the unbiased sample estimate, how is the sampling error defined?
Sampling error is defined as the absolute difference between the sample statistic (the estimator) and the true population parameter it is intended to estimate. In this case, it is the absolute difference between the sample mean x̅ and the population mean μ.
25
Which term refers to the standard deviation of a sampling distribution of a statistic?
The standard error is defined as the standard deviation of the sampling distribution of a statistic, such as the sample mean. It quantifies how much the sample statistic is expected to fluctuate from one sample to another, providing a measure of the precision of the estimate.
26
What is the standard deviation of a sampling distribution commonly referred to as?
The standard error is the standard deviation of the sampling distribution of a statistic. It measures the amount of variability in the sample statistic from sample to sample. A smaller standard error indicates that the sample statistic is a more precise estimate of the population parameter.
27
Regardless of the distribution of the sample or the population, what must the mean of the sampling distribution of the sample mean equal?
According to the properties of sampling distributions, the expected value of the sample mean is equal to the population mean (E[x̅] = μ), making the sample mean an unbiased estimator of the population mean.
28
What is the formal definition for the probability distribution of a sample statistic?
A sampling distribution is the probability distribution of a given statistic based on all possible random samples of a fixed size from a population. It is a fundamental concept in inferential statistics, allowing researchers to calculate the likelihood of observing specific sample results.
29
What is the formula for the standard deviation (standard error) of a sample proportion in a sampling distribution?
The standard error of the sample proportion, denoted as σₚ, measures the dispersion of the sample proportion around the population proportion. It is calculated as the square root of [p(1-p) / n], where p is the population proportion and n is the sample size. Since q = 1-p, this is often written as the square root of (pq / n).
30
Calculate the standard deviation of the sample proportion given a population proportion p = 0.70 and a sample size n = 28.
The standard deviation of a sample proportion is calculated using the formula sqrt(p(1-p)/n). Substituting the given values: sqrt(0.70 * 0.30 / 28) = sqrt(0.21 / 28) = sqrt(0.0075) which is approximately 0.0866. Rounding to three decimal places, we get 0.087.
