Answered

Each year, the newspaper America at a Glance writes an article about the performance of students on a particular nationwide math test. Included in the article this year was the following frequency distribution. It summarizes the test scores from a study which looked at 49 students who completed preparation programs for the test.

\begin{tabular}{|c|c|}
\hline
Mathematics Test Score & Frequency \\
\hline
550 to 599 & 6 \\
600 to 649 & 10 \\
650 to 699 & 14 \\
700 to 749 & 11 \\
750 to 799 & 8 \\
\hline
\end{tabular}

Based on the frequency distribution, using the midpoint of each data class, estimate the mean mathematics test score for the students in the study. For your intermediate computations, use four or more decimal places, and round your answer to one decimal place.

[tex]$\square$[/tex]



Answer :

GinnyAnswer
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].