Final answer:
An increase in standard deviation requires a larger sample size to maintain the same level of precision and confidence in statistical estimates, due to the increased variability in the data.
Step-by-step explanation:
When considering the effect of an increase in the standard deviation on the required sample size for mean-per-unit estimation and probability-proportional-to-size sampling, it is important to note that the sample size will increase. This is because a larger standard deviation implies greater variability within the data, which in turn requires a larger sample to achieve the same level of precision and confidence in estimates. As the standard deviation is a measure of dispersion, a higher value indicates that data points are spread out more widely from the mean, thereby increasing the sampling variability.
With no change in desired precision and confidence, we must adjust the sample size to maintain the same level of accuracy in our estimations. Larger sample sizes are associated with smaller error bounds, leading to a more precise confidence interval. Hence, if only the standard deviation increases, without alterations to other factors, statisticians will need a larger sample size to compensate for the increased variability and to preserve the required precision and confidence.