Question

the following data represent the muzzle velocity (in feet per second) of rounds fired from a 155 - mm gun. for each round, two measurements of the velocity were recorded using two different measuring devices, resulting in the following data. complete parts (a) through (d) below.
observation 1 2 3 4 5 6
a 792.7 790.8 790.1 791.4 790.8 794.5
b 801.4 789.0 799.9 790.0 795.0 791.1
(a) draw a boxplot of the differenced data. does this visual evidence support the results obtained in part (b)?
which boxplot below represents the data?
a.
-12 -10 -8 -6 -4 -2 0 2 4
differences
b.
-12 -10 -8 -6 -4 -2 0 2 4
differences
c.
-12 -10 -8 -6 -4 -2 0 2 4
differences
d.
-12 -10 -8 -6 -4 -2 0 2 4
differences

Explanation:

Step1: Calculate Differences

First, find the differences \( d = A - B \) for each observation:

• Observation 1: \( 792.7 - 801.4 = -8.7 \)
• Observation 2: \( 790.8 - 789.0 = 1.8 \)
• Observation 3: \( 790.1 - 799.9 = -9.8 \)
• Observation 4: \( 791.4 - 790.0 = 1.4 \)
• Observation 5: \( 790.8 - 795.0 = -4.2 \)
• Observation 6: \( 794.5 - 791.1 = 3.4 \)

So the differences are: \(-8.7, 1.8, -9.8, 1.4, -4.2, 3.4\)

Step2: Analyze Boxplot Features

A boxplot shows the median, quartiles, and outliers. Let's find key values:

• Sort the differences: \(-9.8, -8.7, -4.2, 1.4, 1.8, 3.4\)
• Median (Q2): Average of 3rd and 4th values: \(\frac{-4.2 + 1.4}{2} = -1.4\)
• Q1 (median of lower half \(-9.8, -8.7, -4.2\)): \(-8.7\)
• Q3 (median of upper half \(1.4, 1.8, 3.4\)): \(1.8\)
• Minimum: \(-9.8\), Maximum: \(3.4\)

Now, check the boxplots. The spread and median should match. The differences have a mix of negative and positive, with lower quartile around -8.7, median around -1.4, upper quartile around 1.8. The boxplot that aligns with these values (looking at the scale and box position) is likely the one with the correct spread. From the options, the boxplot that matches the range and quartiles (e.g., minimum ~-10, maximum ~4, median around -1 to 0) is option A (or the one with the box positioned to reflect the median and quartiles we calculated). Wait, re - checking the differences: when we sort \(-9.8, -8.7, -4.2, 1.4, 1.8, 3.4\), the lower half is \(-9.8, -8.7, -4.2\) (Q1 is -8.7), upper half \(1.4, 1.8, 3.4\) (Q3 is 1.8), median -1.4. The boxplot with the box from ~-8.7 to ~1.8, median around -1.4, and whiskers to -9.8 and 3.4. Looking at the options, the first boxplot (A) seems to have the correct box position. Wait, maybe I made a mistake in the initial calculation. Wait, no, let's recalculate differences:

Wait, Observation 1: 792.7 - 801.4 = -8.7 (correct)

Observation 2: 790.8 - 789.0 = 1.8 (correct)

Observation 3: 790.1 - 799.9 = -9.8 (correct)

Observation 4: 791.4 - 790.0 = 1.4 (correct)

Observation 5: 790.8 - 795.0 = -4.2 (correct)

Observation 6: 794.5 - 791.1 = 3.4 (correct)

Now, let's find the five - number summary:

Minimum: -9.8

Q1: The median of the first three numbers (-9.8, -8.7, -4.2) is -8.7

Median (Q2): (-4.2 + 1.4)/2=-1.4

Q3: The median of (1.4, 1.8, 3.4) is 1.8

Maximum: 3.4

Now, looking at the boxplots, the one with the box from Q1 (-8.7) to Q3 (1.8), median around -1.4, whiskers to -9.8 and 3.4. The first boxplot (Option A) has the box in a position that matches this (the x - axis is from -12 to 4, so -9.8 is near -10, 3.4 near 3 - 4, Q1 ~-8.7, Q3 ~1.8, median ~-1.4). So the correct boxplot is the one labeled A (assuming the first option is A as per the image labels).

Answer:

A. (the boxplot labeled A with the differences axis from -12 to 4, box reflecting the quartiles and median we calculated)