To determine the minimum sample size needed to be 95% confident that the sample standard deviation [tex]\( s \)[/tex] is within 40% of the population standard deviation [tex]\( \sigma \)[/tex], we can refer directly to the given table.
Let's break down the steps:
1. Identify the Confidence Level: The question specifies a 95% confidence level.
2. Identify the Desired Accuracy: We need the sample standard deviation to be within 40% of the population standard deviation, [tex]\( \sigma \)[/tex].
3. Refer to the Table: We find the section of the table corresponding to a 95% confidence level and look in the column for 40%.
The table provides the necessary sample sizes for different confidence levels and desired percentages of the population standard deviation:
[tex]\[
\begin{array}{|l|c|c|c|c|c|c|c|}
\hline
\text{To be 95\% confident that s is within} & 1\% & 5\% & 10\% & 20\% & 30\% & 40\% & 50\% \\
\hline
\text{of the value of } \sigma, \text{ the sample size } n \text{ should be at least} & 19.205 & 768 & 192 & 48 & 21 & 12 & \\
\hline
\end{array}
\][/tex]
From the table, when we focus on the 95% confidence level row and the 40% column, we see that the minimum sample size [tex]\( n \)[/tex] should be at least 12.
Therefore, the minimum sample size needed is [tex]\( \boxed{12} \)[/tex].
