consider the boxplot below.\n43 58 61 67 74\na. what quarter has the smallest spread of data?\nfirst\nfourth\…

Explanation:

Step1: Recall box - plot properties

In a box - plot, the box is divided into two parts by the median. The lower part of the box represents the second quarter (from the first quartile $Q_1$ to the median $Q_2$) and the upper part of the box represents the third quarter (from the median $Q_2$ to the third quartile $Q_3$). The whiskers represent the first and fourth quarters. The spread of a quarter is the difference between the upper and lower values of that quarter.

Step2: Calculate spreads for each quarter

The first quarter has values from $43$ to $58$, so the spread of the first quarter is $58 - 43=15$. The second quarter has values from $58$ to $61$, so the spread of the second quarter is $61 - 58 = 3$. The third quarter has values from $61$ to $67$, so the spread of the third quarter is $67 - 61=6$. The fourth quarter has values from $67$ to $74$, so the spread of the fourth quarter is $74 - 67 = 7$.

Step3: Answer part a

The second quarter has the smallest spread of data.

Step4: Answer part b

The spread of the second - quarter is $61 - 58=3$.

Step5: Answer part c

The first quarter has the largest spread of data.

Step6: Answer part d

The spread of the first - quarter is $58 - 43 = 15$.

Step7: Recall IQR formula

The inter - quartile range (IQR) is defined as $IQR=Q_3 - Q_1$. Here, $Q_1 = 58$ and $Q_3 = 67$, so $IQR=67 - 58 = 9$.

Step8: Recall data concentration in box - plot

The box of the box - plot (from $Q_1$ to $Q_3$) contains the middle 50% of the data. The interval $58 - 67$ (which is the box) has the most data in it. Among the given options, the interval $57 - 61$ is part of the box and is the most data - dense among them.

Answer:

a. Second b. 3 c. First d. 15 e. 9 f. 57 - 61