\begin{tabular}{|c|c|}
\hline
1 & [tex]$1 J$[/tex] \\
\hline
2 & 14.45 \\
\hline
3 & 14.91 \\
\hline
4 & 15.09 \\
\hline
5 & 14.91 \\
\hline
6 & 15.82 \\
\hline
7 & 14.36 \\
\hline
8 & 15.55 \\
\hline
9 & 14.36 \\
\hline
10 & 15.27 \\
\hline
\end{tabular}

Use this information to complete the statements.

The average of the sample means is [tex]$\square$[/tex].

As the number of surveys conducted increases, the average of the sample means approaches the [tex]$\square$[/tex].



Answer :

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]