To determine the minimum sample size needed to be 99% confident that the sample standard deviation [tex]\( s \)[/tex] is within 50% of [tex]\( \sigma \)[/tex], we can refer to the provided table. The table consists of various levels of confidence and margins of error for the sample standard deviation [tex]\( s \)[/tex] in relation to the population standard deviation [tex]\( \sigma \)[/tex].
Let's carefully interpret the table:
1. The first section of the table details the sample sizes needed to achieve a specified level of confidence (95%) for different error margins.
2. The second section of the table details the sample sizes needed to achieve a specified level of confidence (99%) for various error margins.
We are particularly interested in the second section, where the confidence level is 99%.
From the table:
- For a 99% confidence level and an error margin within 50% of [tex]\( \sigma \)[/tex], the minimum sample size needed is given in the last column under the row for 50% margin.
Based on the second section of the table, the necessary sample size for a 99% confidence level and a margin of error within 50% of [tex]\( \sigma \)[/tex] is [tex]\( 14 \)[/tex].
Thus, the minimum sample size needed is [tex]\( \boxed{14} \)[/tex].
Is this sample size practical?
Considering the practicality of obtaining a sample size of 14:
- A sample size of 14 is relatively small and should be practical for most statistical analysis scenarios, especially when working with a normally distributed population. Typically, smaller sample sizes are easier and less costly to collect, making this sample size quite feasible in practice.
