Answer:
a) The value of the median is 112
b) The value of the lower quartile is 95.59
Step-by-step explanation:
From the histogram, we look for the data values as follows
Score, Frequency, Cumulative , n
80 - 90, 2.6, 2.6
90 - 100, 3.4, 6
100 - 120, 5, 11
120 - 140, 3, 14
140 - 170, 2.4, 16.4
170 - 200, 0.6, 17
The median is therefore the (nth + 1)/2th value = (17 + 1)/2 = 9th value
The 9th value is between the 100 - 120 score values
We note that the previous n value before the 100 - 120 range = 6, therefore, the 9th value will be the weighted average in the 100 - 120 range of tha amount the 9th value is larger than the 6th value as follows;
9 - 6 = 3
The weighted average of 3 out 5 (11 - 6 = 5) of the 100 - 120 range which gives;
Therefore, the value of the median, Q₂ = 100 + (9 - 6)/(11 - 6))*(120 - 100) = 112
b) The lower quartile = Q₁ is given by the value at the (n +1)/4th position = (17 + 1)/4 = 4.5th position
From the cumulative frequency column, the 4.5th value is in the 90 - 100 score range
Similar to the previous question, we find the value of the lower quartile using the weighted average as follows;
Therefore, the value of the lower quartile, Q₁ = 90 + (4.5 - 2.6)/(6 - 2.6)×(100 - 90) = 95.59.
