Answer :
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].
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].
