To find the third quartile ([tex]\(Q_3\)[/tex]) of the given test scores, follow these steps:
1. List the Scores in Ascending Order:
The scores are already provided in ascending order:
[tex]\[
32, 37, 41, 44, 46, 48, 53, 55, 56, 57, 59, 63, 65, 66, 68, 69,
70, 71, 74, 74, 75, 77, 78, 79, 80, 82, 83, 86, 89, 92, 95, 99
\][/tex]
2. Determine the Position of the Third Quartile [tex]\(Q_3\)[/tex]:
The third quartile ([tex]\(Q_3\)[/tex]) is the median of the upper half of the dataset. In a dataset with [tex]\( n \)[/tex] values, [tex]\(Q_3\)[/tex] is located at the position [tex]\( \frac{3(n+1)}{4} \)[/tex].
For [tex]\(n = 32\)[/tex]:
[tex]\[
\text{Position of } Q_3 = \frac{3(32+1)}{4} = \frac{3 \times 33}{4} = \frac{99}{4} = 24.75
\][/tex]
This means [tex]\(Q_3\)[/tex] is located between the 24th and 25th values in the dataset.
3. Find the Values at the 24th and 25th Positions:
Look at the scores list:
[tex]\[
\text{24th value = } 79, \quad \text{25th value = } 80
\][/tex]
4. Calculate the Third Quartile [tex]\(Q_3\)[/tex]:
Since the position is 24.75, [tex]\(Q_3\)[/tex] is three-quarters of the way between the 24th and 25th values. We need to perform a linear interpolation:
[tex]\[
Q_3 = 79 + 0.75 (80 - 79) = 79 + 0.75 \times 1 = 79 + 0.75 = 79.25
\][/tex]
Therefore, the third quartile ([tex]\(Q_3\)[/tex]) of the given test scores is [tex]\(79.25\)[/tex].
