Consider the following partially completed frequency table:
Class xi fi
[25; 30) 27.5 1
[30; 35) 32.5 3
[35; 40) 37.5 9
[40; 45) 42.5 4
[45; 50) 47.5 3
Determine the third quartile by using the appropriate formula.
35.5
25.5
36.5
42.5
To determine the third quartile, we need to find the position of the 75th percentile in the data set. Since the data is grouped in intervals, we can use the formula for finding the quartiles in grouped data:
Q3 = L + [(N/4 - F)/f] * w
Where:
Q3 = Third quartile
L = Lower boundary of the interval containing the 75th percentile
N = Total number of data points
F = The cumulative frequency of the interval containing the 75th percentile
f = Frequency of the interval containing the 75th percentile
w = Width of the interval
In this case:
N = 20 (sum of all frequencies)
The 75th percentile is at the 15th data point (20 * 0.75 = 15)
So, calculating the third quartile:
Q3 = 35 + [(15 - 4)/9] * 5
Q3 = 35 + (11/9) * 5
Q3 = 35 + 5.5556
Q3 = 40.5556
Therefore, the third quartile is approximately 40.56, which falls within the [40, 45) interval, making it 42.5.
So, the correct answer is 42.5.