Sure, I'd be happy to explain this step by step.
Given the data from the table:
[tex]\[
\begin{array}{|c|c|}
\hline
1 & 14.45 \\
\hline
2 & 14.91 \\
\hline
3 & 15.09 \\
\hline
4 & 14.91 \\
\hline
5 & 15.82 \\
\hline
6 & 14.36 \\
\hline
7 & 15.55 \\
\hline
8 & 14.36 \\
\hline
9 & 15.27 \\
\hline
\end{array}
\][/tex]
Step-by-step solution:
1. List the Sample Means:
The sample means provided are:
[tex]\[ 14.45, 14.91, 15.09, 14.91, 15.82, 14.36, 15.55, 14.36, 15.27 \][/tex]
2. Calculate the Average of the Sample Means:
To find the average, sum up all the sample means and then divide by the number of sample means.
[tex]\[
\text{Sum of sample means} = 14.45 + 14.91 + 15.09 + 14.91 + 15.82 + 14.36 + 15.55 + 14.36 + 15.27
\][/tex]
[tex]\[
\text{Sum} = 14.45 + 14.91 + 15.09 + 14.91 + 15.82 + 14.36 + 15.55 + 14.36 + 15.27 = 134.72
\][/tex]
[tex]\[
\text{Number of sample means} = 9
\][/tex]
[tex]\[
\text{Average} = \frac{134.72}{9} \approx 14.96888888888889
\][/tex]
Rounded to two decimal places, the average is approximately [tex]\(14.97\)[/tex].
Therefore:
[tex]\[
\text{The average of the sample means is } 14.97
\][/tex]
3. General Principle of Increasing Sample Sizes:
As the number of surveys conducted increases, the average of the sample means tends to approach the population mean. This is a principle from statistics known as the Law of Large Numbers.
Final Statements:
1. The average of the sample means is [tex]\( \boxed{14.97} \)[/tex]
2. As the number of surveys conducted increases, the average of the sample means approaches the [tex]\( \boxed{\text{population mean}} \)[/tex]
