Assume that the sample is a simple random sample obtained from a normally distributed population of flight delays at an airport. Use the table below to find the minimum sample size needed to be [tex]$95\%$[/tex] confident that the sample standard deviation is within [tex]$40\%$[/tex] of the population standard deviation. A histogram of a sample of those arrival delays suggests the distribution is skewed, not normal. How does the distribution affect the sample size?

\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
\begin{tabular}{l}
To be [tex]$95\%$[/tex] confident that [tex]$s$[/tex] is \\
within
\end{tabular} & [tex]$1\%$[/tex] & [tex]$5\%$[/tex] & [tex]$10\%$[/tex] & [tex]$20\%$[/tex] & [tex]$30\%$[/tex] & [tex]$40\%$[/tex] & [tex]$50\%$[/tex] \\
\hline
\begin{tabular}{l}
of the value of [tex]$\sigma$[/tex], the sample size \\
[tex]$n$[/tex] should be at least
\end{tabular} & 19,205 & 768 & 192 & 48 & 21 & 12 & 7 \\
\hline
\end{tabular}

The minimum sample size needed is [tex]$\square$[/tex]



Answer :

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].