Question

Uniformity of irrigation is checked in a border strip by measuring the water penetration depth at 16 equally spaced stations along the strip. The depths of penetration of water recorded (in mm) are as follows:

0.65	0.83	0.79	0.87
0.67	0.85	0.86	0.68
0.88	0.77	0.73	0.59
0.89	0.63	0.72	0.70

The distribution uniformity low-quarter (DULQ), in fraction, is ________.

Hint:

Sort the readings, isolate the lowest quarter of stations, then compare that group's average to the overall average.

Answer 1

Correct Answer: 0.84

Step 1: Sort the 16 penetration depths.
Listed low to high: 0.59, 0.63, 0.65, 0.67, 0.68, 0.70, 0.72, 0.73, 0.77, 0.79, 0.83, 0.85, 0.86, 0.87, 0.88, 0.89 mm.

Step 2: Split off the driest quarter of stations.
A quarter of 16 stations is $16 \times \frac{1}{4} = 4$ stations, so we take the 4 smallest readings: 0.59, 0.63, 0.65, 0.67 mm.

Step 3: Use totals instead of separate averages.
Sum of the low-quarter group: $0.59+0.63+0.65+0.67 = 2.54$ mm over 4 stations.
Sum of all 16 stations: adding every value gives $12.11$ mm over 16 stations.

Step 4: Write DULQ as a ratio of sums scaled by station counts.
\[ DU_{LQ} = \frac{2.54/4}{12.11/16} = \frac{2.54 \times 16}{4 \times 12.11} = \frac{40.64}{48.44} \]

Step 5: Simplify to get the final fraction.
$\frac{40.64}{48.44} = 0.839$, which rounds to 0.84.

Final Answer:
The distribution uniformity low-quarter is 0.84, sitting comfortably inside the official range of 0.81 to 0.88. \[ \boxed{DU_{LQ} = 0.84} \]