To estimate the mean mathematics test score for the students in the study based on the given frequency distribution, we will follow these steps:
1. Identify the class intervals and their frequencies:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Class Interval} & \text{Frequency} \\
\hline
550 - 599 & 6 \\
600 - 649 & 10 \\
650 - 699 & 14 \\
700 - 749 & 11 \\
750 - 799 & 8 \\
\hline
\end{array}
\][/tex]
2. Calculate the midpoints of each class interval:
The midpoint of a class interval is calculated as:
[tex]\[
\text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}
\][/tex]
Using this formula, we get the midpoints for each class interval:
[tex]\[
\begin{align*}
\text{Midpoint of } 550 - 599 & = \frac{550 + 599}{2} = 574.5 \\
\text{Midpoint of } 600 - 649 & = \frac{600 + 649}{2} = 624.5 \\
\text{Midpoint of } 650 - 699 & = \frac{650 + 699}{2} = 674.5 \\
\text{Midpoint of } 700 - 749 & = \frac{700 + 749}{2} = 724.5 \\
\text{Midpoint of } 750 - 799 & = \frac{750 + 799}{2} = 774.5 \\
\end{align*}
\][/tex]
3. Calculate the product of each midpoint and its respective frequency:
[tex]\[
\begin{align*}
574.5 \times 6 & = 3447.0 \\
624.5 \times 10 & = 6245.0 \\
674.5 \times 14 & = 9443.0 \\
724.5 \times 11 & = 7969.5 \\
774.5 \times 8 & = 6196.0 \\
\end{align*}
\][/tex]
4. Sum these products:
[tex]\[
3447.0 + 6245.0 + 9443.0 + 7969.5 + 6196.0 = 33300.5
\][/tex]
5. Calculate the total frequency:
[tex]\[
6 + 10 + 14 + 11 + 8 = 49
\][/tex]
6. Estimate the mean mathematics test score:
The estimated mean is given by:
[tex]\[
\text{Estimated Mean} = \frac{\text{Sum of products of midpoints and frequencies}}{\text{Total frequency}}
\][/tex]
Substituting the values we have:
[tex]\[
\text{Estimated Mean} = \frac{33300.5}{49} = 679.6
\][/tex]
Thus, the estimated mean mathematics test score for the students in the study is [tex]\( \boxed{679.6} \)[/tex].
