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Assume that the sample is a simple random sample obtained from a normally distributed population of IQ scores of statistics professors. Use the table below to find the minimum sample size needed. Is this sample size practical?

\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 & 8 \\
\hline
\begin{tabular}{l}
To be [tex]$99 \%$[/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
of the value of [tex]$\sigma$[/tex], the sample size [tex]$n$[/tex] should be at least \\
\hline
\end{tabular}

The minimum sample size needed is 33,218. Is this sample size practical?

A. No, because the sample size is excessively large to be practical for most applications.
B. Yes, because the sample size should be as large as possible for most applications.
C. Yes, because the sample size is small enough to be practical for most applications.



Answer :

GinnyAnswer
To determine whether the sample size of 33,218 is practical, let's first consider the context and the provided information.

The table presents the minimum sample sizes required to estimate the population standard deviation (σ) within various percentages of its value (1%, 5%, 10%, 20%, 30%, 40%, and 50%) with a certain level of confidence (either 95% or 99%).

Here, we are asked about the practicality of a sample size of 33,218. Let's look at the general trends in the table:
- For a 95% confidence level, to estimate σ within just 1% of its value, a sample size of 19,205 is mentioned.
- For a 99% confidence level with high accuracy (within 1%), the sample size needed would logically be even higher than 19,205, although the exact figure is not provided in the table.

Understanding these data points, let's analyze the given options:

Option A: No, because the sample size is excessively large to be practical for most applications.
Given that typical sample size recommendations in your table max out at 19,205 for estimating σ within 1% with 95% confidence, a required sample of 33,218 is significantly larger. This suggests that such a large sample size would be impractical or excessively large for most real-world applications, especially considering factors like time, cost, and resource limitations.

Option B: Yes, because the sample size should be as large as possible for most applications.
While larger sample sizes generally provide better estimates and more precise results, practical constraints often limit the feasibility of obtaining very large samples. Hence, this statement is not practical.

Option C: Yes, because the sample size is small enough to be practical for most applications.
This option is clearly false since 33,218 is not a small sample size by any measure, especially compared to the table's range of sample sizes.

Consequently, based on the analysis of the required sample sizes and considerations of practicality, the correct answer is:
A. No, because the sample size is excessively large to be practical for most applications.

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