- The standard error of the mean formula is \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\).
- Since \(n \ge 1\), then \(\sqrt{n} \ge 1\), which implies \(\sigma_{\bar{x}} \le \sigma\).
- Statement (d) claims the standard error is larger than the population standard deviation, which is mathematically impossible.
- The false statement is \(\boxed{d}\).
Explanation
- Analyze the Standard Error formula and its components
The standard error of the mean (SEM), denoted by \(\sigma_{\bar{x}}\), is a fundamental concept in statistics that measures how much the sample mean of a population is expected to deviate from the true population mean. It is defined by the formula:
\[\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\]
where \(\sigma\) is the standard deviation of the population and \(n\) is the sample size. We need to evaluate each of the four given statements to determine which one is false.
- Evaluate each given statement
- Statement (a): "it decreases as the sample size increases." Looking at the formula \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\), we see that the sample size \(n\) is in the denominator. As \(n\) increases, \(\sqrt{n}\) increases, which means the value of the fraction \(\frac{\sigma}{\sqrt{n}}\) decreases. Thus, this statement is true.
- Statement (b): "its value is influenced by the standard deviation of the population." The formula shows that \(\sigma_{\bar{x}}\) is directly proportional to \(\sigma\). If the population variability changes, the standard error changes. Thus, this statement is true.
- Statement (c): "it measures the variability in sample means." By definition, the standard error of the mean represents the standard deviation of the sampling distribution of the sample mean. It quantifies how much sample means are expected to fluctuate from sample to sample. Thus, this statement is true.
- Statement (d): "it is larger than the standard deviation of the population." Let's compare \(\sigma_{\bar{x}}\) to \(\sigma\). Since \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\) and for any sample size \(n \ge 1\), the term \(\sqrt{n} \ge 1\), it follows that \(\sigma_{\bar{x}} \le \sigma\). Specifically, if \(n > 1\), then \(\sqrt{n} > 1\), which makes \(\sigma_{\bar{x}} < \sigma\). Therefore, the standard error can never be larger than the standard deviation of the population. Thus, this statement is false.
- Conclusion regarding the false statement
Based on the analysis, statements (a), (b), and (c) are all accurate descriptions of the properties of the standard error of the mean. Statement (d) claims that the standard error is larger than the population standard deviation, which contradicts the mathematical relationship \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\) where \(n \ge 1\). Therefore, statement (d) is the incorrect one.
Examples
In real-world data analysis, researchers use the standard error to construct confidence intervals around a sample mean. For example, if you measure the heights of students in a school, the standard error helps you determine how precise your average height estimate is; a smaller standard error (achieved with a larger sample size) means your estimate is likely closer to the true average height of all students in the school. By understanding that the standard error is always less than or equal to the population standard deviation, researchers maintain realistic expectations about the precision of their findings.