Answered

The side-by-side stemplot below displays the arm spans in centimeters, for two classes.




\multicolumn{3}{|c|}{Arm Span (cm)}

Class A		Class B

8		14

31		153

965		1556

421		16022233

995		16567

10		1703

9965		17

320		18012

66		189

0		192

19

202

\multicolumn{4}{|c|}{Key: $15 \mid 5=155 cm$}






Which statement correctly compares the variability of the arm spans for Class A to that of Class B?


AThe arm spans for Class A have more variability than the arm spans for Class B.
BThe arm spans for Class B have less variability than the arm spans for Class A.
CThe arm spans for Class A have less variability than the arm spans for Class B.
DThe arm spans for Class B have about the same variability as the arm spans for Class A.

Verified Answer :

  • Identify the minimum and maximum for Class A as \(148\) and \(190\), resulting in a range of \(42\).
  • Identify the minimum and maximum for Class B as \(153\) and \(202\), resulting in a range of \(49\).
  • Compare the two ranges, noting that \(42 < 49\).
  • The arm spans for Class A have less variability than the arm spans for Class B.
  • Final Answer: \(\boxed{\text{The arm spans for Class A have less variability than the arm spans for Class B.}}\)

Explanation

  1. Identify minimum and maximum values

To compare the variability of the arm spans between Class A and Class B, we first determine the range of each dataset. The range is calculated as the difference between the maximum and minimum values in the set, providing a simple measure of the total spread of the data.

From the provided stemplot:

  • For Class A, the smallest value (minimum) is found in the stem 14 row, which is \(148\) cm. The largest value (maximum) is found in the stem 19 row, which is \(190\) cm.
  • For Class B, the smallest value (minimum) is found in the stem 15 row, which is \(153\) cm. The largest value (maximum) is found in the stem 20 row, which is \(202\) cm.
  1. Calculate the ranges

Using the identified minimum and maximum values, we calculate the range for each class:

  • For Class A:

\(\text{Range}_A = \text{max}_A - \text{min}_A = 190 - 148 = 42\) cm.

  • For Class B:

\(\text{Range}_B = \text{max}_B - \text{min}_B = 202 - 153 = 49\) cm.

  1. Compare the variability and conclude

Comparing the two ranges:

  • \(\text{Range}_A = 42\) cm
  • \(\text{Range}_B = 49\) cm

Since \(\text{Range}_B > \text{Range}_A\), the arm spans for Class B cover a wider interval than those for Class A. Therefore, the arm spans for Class A have less variability than the arm spans for Class B.

Examples

In real-world data analysis, comparing variability is crucial. For instance, if you're a coach comparing the physical development of two groups of athletes, calculating the range or interquartile range of their arm spans or heights helps you understand the diversity of physical traits in your teams. Knowing that one group has more variability (\(\text{Range}_B = 49\) cm vs \(\text{Range}_A = 42\) cm) allows you to tailor training plans more effectively to each group's unique distribution of physical characteristics.

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