A random sample of 10 economists produced the following...

A random sample of 10 economists produced the following forecasts for percentage growth in real gross domestic product in the next year: $\begin{array}{llllllllll}2.2 & 2.8 & 3.0 & 2.5 & 2.4 & 2.6 & 2.5 & 2.4 & 2.7 & 2.6\end{array}$ Use unbiased estimation procedures to find point estimates for the following: a. The population mean b. The population variance c. The variance of the sample mean d. The population proportion of economists predicting growth of at least $2.5 \%$ in real gross domestic product

A random sample of 10 economists produced the following forecasts for percentage growth in real gross domestic product in the next year:
$\begin{array}{llllllllll}2.2 & 2.8 & 3.0 & 2.5 & 2.4 & 2.6 & 2.5 & 2.4 & 2.7 & 2.6\end{array}$
Use unbiased estimation procedures to find point estimates for the following:
a. The population mean
b. The population variance
c. The variance of the sample mean
d. The population proportion of economists predicting growth of at least $2.5 \%$ in real gross domestic product

Key Concepts

Point Estimation

Point estimation involves using sample data to calculate a single value, or statistic, that serves as the best estimate of an unknown parameter of the population. In statistics, point estimators provide a concise summary, such as an average or proportion, to summarize one aspect of the population distribution.

Population Mean

The population mean is the average of all values in the entire population. In many practical scenarios, the sample mean is used as an estimator for the population mean because it is an unbiased estimator, meaning its mean equals the true population mean.

Population Variance

Population variance measures the degree to which individual values in the population differ from the population mean. When estimating the population variance using sample data, the unbiased estimator is typically calculated by dividing the sum of squared deviations from the sample mean by n-1, rather than n.

Variance of the Sample Mean

The variance of the sample mean, often known as the standard error squared, quantifies the variability of the sample mean from sample to sample. It is estimated by dividing the sample variance by the sample size, reflecting how the estimate of the population mean stabilizes as the sample size increases.

Population Proportion Estimation

Population proportion estimation involves calculating the proportion of individuals in a population that have a specific attribute. When using sample data, the sample proportion is employed as a point estimator, providing an unbiased estimate of the true population proportion.

Unbiased Estimation

Unbiased estimation refers to a property of an estimator where the expected value of the estimator is equal to the true parameter value. In other words, an unbiased estimator does not systematically overestimate or underestimate the parameter it is designed to estimate.

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