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The mean game scores and standard deviations of four seasons of a football team are given below.

\begin{tabular}{|c|c|c|}
\hline
Season & Mean & Standard Deviation \\
\hline
2005 & 19 & 3.5 \\
\hline
2006 & 21 & 2.8 \\
\hline
2007 & 12 & 1.0 \\
\hline
2008 & 9 & 4.0 \\
\hline
\end{tabular}

Which statement must be true?

A. The game scores of season 2006 have a greater spread than the game scores of season 2008.

B. The game scores of season 2007 have a greater spread than the game scores of season 2005.

C. The game scores of season 2006 have a greater spread than the game scores of season 2005.

D. The game scores of season 2005 have a greater spread than the game scores of season 2006.



Answer :

GinnyAnswer
Let's analyze the standard deviations provided in the question to determine which statement is true:

1. Season 2005: Standard Deviation = 3.5
2. Season 2006: Standard Deviation = 2.8
3. Season 2007: Standard Deviation = 1.0
4. Season 2008: Standard Deviation = 4.0

Review the statements one by one in terms of standard deviations:

1. The game scores of season 2006 have a greater spread than the game scores of season 2008.
- Comparing standard deviations: \( 2.8 \) (2006) vs \( 4.0 \) (2008)
- Since \( 2.8 < 4.0 \), this statement is false.

2. The game scores of season 2007 have a greater spread than the game scores of season 2005.
- Comparing standard deviations: \( 1.0 \) (2007) vs \( 3.5 \) (2005)
- Since \( 1.0 < 3.5 \), this statement is false.

3. The game scores of season 2006 have a greater spread than the game scores of season 2005.
- Comparing standard deviations: \( 2.8 \) (2006) vs \( 3.5 \) (2005)
- Since \( 2.8 < 3.5 \), this statement is false.

4. The game scores of season 2005 have a greater spread than the game scores of season 2006.
- Comparing standard deviations: \( 3.5 \) (2005) vs \( 2.8 \) (2006)
- Since \( 3.5 > 2.8 \), this statement is true.

Therefore, the only true statement is:

The game scores of season 2005 have a greater spread than the game scores of season 2006.

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