Final answer:
The correct answer is option A. Increases in the estimated standard deviation lead to an increase in the required sample size for mean-per-unit samples because a larger sample size is needed to maintain a desired level of precision in estimates.
Step-by-step explanation:
Increases in the estimated standard deviation have the effect of increasing the required sample size in a mean-per-unit sample. When estimating a population mean, statisticians use a sample mean to infer about the population. The precision of this estimation is quantified by the standard deviation of the sampling distribution of sample means, often referred to as the standard error. As the estimated standard deviation of the population increases, the variability of sample means also increases, implying that a larger sample size will be necessary to achieve a desired level of precision.
The relationship between sample size and the standard deviation can be understood through the formula for calculating the standard error of the mean: SE = SD/√n, where SE is the standard error, SD is the standard deviation, and n is the sample size. If the standard deviation (SD) increases, the standard error (SE) will also increase unless the sample size (n) is increased to compensate. Therefore, a larger sample size is needed to ensure that the standard error remains small to provide precise estimates.
Moreover, this principle is central to hypothesis testing and confidence interval estimations. For instance, when constructing a confidence interval for a mean, a larger standard deviation results in a wider interval, assuming the sample size remains the same. To narrow the confidence interval to a level that is deemed acceptable, increasing the sample size is essential.