Question

Which of the following combinations guarantees a larger sample size?

1) Decrease the desired confidence level and decrease the tolerable deviation rate.
2) Increase the desired confidence level and decrease the tolerable deviation rate.
3) Decrease the desired confidence level and increase the expected deviation rate.
4) Increase the tolerable deviation rate and increase the expected deviation rate.

Answer 1

The combination that guarantees a larger sample size is to increase the desired confidence level and decrease the tolerable deviation rate, as both of these individually call for a larger sample size to maintain the specified precision and confidence in statistical estimates.

Step-by-step explanation:

To determine which combination guarantees a larger sample size, we must consider the impact of different factors on required sample size for a statistical study.

• Increasing the desired confidence level leads to a higher critical value, which results in a wider confidence interval and a larger sample size needed to maintain the same error bound.
• Decreasing the tolerable deviation rate (also known as the error bound or margin of error) requires a larger sample to ensure that the estimates fall within this narrower range with the same level of confidence.
• Increasing the expected deviation rate would imply that there's more variability in the data, which typically calls for a larger sample to accurately estimate the population parameter.

Considering these factors:

1. Decreasing both the confidence level and the tolerable deviation rate could have opposing effects, with the reduced confidence level suggesting a smaller sample size and the decreased tolerable deviation rate suggesting a larger one.
2. Increasing the confidence level and decreasing the tolerable deviation rate would both individually require a larger sample size, so together, they guarantee the need for the largest sample size among the given options.
3. Decreasing the confidence level and increasing the expected deviation rate might result in a need for a smaller sample size because the lower confidence level could offset the increase due to higher expected deviation rate.
4. Increasing both the tolerable deviation rate and the expected deviation rate would typically mean a larger interval and potentially a smaller sample size.

Therefore, the combination that guarantees a larger sample size is the second option: Increase the desired confidence level and decrease the tolerable deviation rate.