Final answer:
The student's statement is incorrect; the standard error of the mean decreases as the sample size increases, not the other way around. This relationship is based on the formula for standard error and is confirmed by the central limit theorem.
Step-by-step explanation:
False. The statement provided by the student is incorrect. The correct relationship is that as the sample size increases, the standard error of the mean decreases. This is because the standard error is inversely proportional to the square root of the sample size (n), which is derived from the formula of the standard error of the mean, SE = σ/√n, where σ is the population standard deviation.
According to the central limit theorem, as the sample size increases, the sampling distribution of the sample means will approximate a normal distribution, regardless of the shape of the population distribution, provided the sample size is sufficiently large. Moreover, the standard deviation of the sampling distribution of the means, also known as the standard error, will decrease, resulting in a more precise estimate of the population mean. As such, the error bound for estimation decreases, hence the confidence interval becomes narrower.
On the contrary, decreasing the sample size would result in an increase in the standard error, thereby making estimates of the population mean less precise and leading to wider confidence intervals.