In a residential neighborhood, the median value of a house...

In a residential neighborhood, the median value of a house is 200,000 . For which of the following sample sizes is the sample median most likely to be above 250,000 ? $\begin{array}{ll}\text { (a) } & n=10\end{array}$ (b) $n=50$ $\begin{array}{ll}\text { (c) } & n=100\end{array}$ (d) $n=1000$ (e) Impossible to determine without more information.

In a residential neighborhood, the median value of a house is 200,000 . For which of the following sample sizes is the sample median most likely to be above 250,000 ?
$\begin{array}{ll}\text { (a) } & n=10\end{array}$
(b) $n=50$
$\begin{array}{ll}\text { (c) } & n=100\end{array}$
(d) $n=1000$
(e) Impossible to determine without more information.

Key Concepts

Effect of Sample Size on Variability

As the sample size increases, the variability or standard error of estimators like the sample median typically decreases. This means that while larger samples provide more precise estimates of the population median, they are less likely to display extreme deviations compared to smaller samples, which can have higher variability.

Sampling Distribution and Variability

The sampling distribution of the sample median describes how the median would vary from one sample to another. With smaller sample sizes, the variability is larger, meaning that the sample median is more likely to deviate from the population median compared to larger samples where the median tends to be more stable.

Sample Median

The sample median is the middle value in a sorted sample of observations, and it serves as an estimator of the population median. It is less sensitive to extreme values compared to the mean, but its accuracy depends on the sample size and the underlying distribution.

Population Median

The population median is the value that splits the entire distribution of a variable such that half the observations fall below it and half above it. It is a robust measure of central tendency that is especially useful when the distribution is skewed or contains outliers.

Order Statistics

Order statistics involve the statistics obtained from the ordered values of a sample. The sample median is a key example of an order statistic, and studying these statistics helps in understanding the variability and reliability of estimates like the sample median.

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