Answered

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




\multirow{2}{*}{Class A}		\multicolumn{3}{|c|}{Arm Span (cm)}

814

31		153

965		1556

421		16022233

995		16567

10		1703

9965		17

320		18O 12

66		189

0		192

19

202

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






Which statement correctly describes the proportion of arm spans that are longer than 170 cm?


AThe proportion of arm spans that are longer than 170 cm is higher in Class B than in Class A.
BThe proportion of arm spans that are longer than 170 cm is higher in Class A than in Class B.
CThe proportion of arm spans that are longer than 170 cm is about the same for Class A and Class B.
DThe proportion of arm spans that are longer than 170 cm cannot be compared with the given information.

Verified Answer :

  • Identify all arm span values: Class A has 23 students and Class B has 21 students.
  • Count values greater than 170 cm: Class A has 11 students, while Class B has 7 students.
  • Calculate the proportions: \(P_A = \frac{11}{23} \approx 0.478\) and \(P_B = \frac{7}{21} \approx 0.333\).
  • Final conclusion: The proportion of arm spans that are longer than 170 cm is higher in Class A than in Class B. \(\boxed{\text{The proportion of arm spans that are longer than 170 cm is higher in Class A than in Class B.}}\)

Explanation

  1. Analyze the data and calculate totals for each class

To determine the proportion of students in each class with an arm span greater than 170 cm, we first need to extract the data for each class from the stemplot. The stemplot lists the stems (hundreds and tens digits) in the middle, and the leaves (ones digit) for Class A on the left and Class B on the right.

  • Class A:
  • Stem 15: 153, 151, 159, 156, 155 (5 students)
  • Stem 16: 164, 162, 161, 169, 169, 165 (6 students)
  • Stem 17: 171, 170, 179, 179, 176, 175 (6 students)
  • Stem 18: 183, 182, 180, 186, 186 (5 students)
  • Stem 19: 190 (1 student)
  • Total students in Class A: \(5 + 6 + 6 + 5 + 1 = 23\).
  • Class B:
  • Stem 14: 148 (1 student)
  • Stem 15: 153, 155, 156 (3 students)
  • Stem 16: 160, 162, 162, 162, 163, 163, 165, 166, 167 (9 students)
  • Stem 17: 170, 173 (2 students)
  • Stem 18: 180, 181, 182, 189 (4 students)
  • Stem 19: 192 (1 student)
  • Stem 20: 202 (1 student)
  • Total students in Class B: \(1 + 3 + 9 + 2 + 4 + 1 + 1 = 21\).
  1. Identify arm spans greater than 170 cm and calculate proportions

Now, let us identify the number of students with arm spans strictly greater than 170 cm.

  • Class A:
  • Values > 170: 171, 179, 179, 176, 175, 183, 182, 180, 186, 186, 190.
  • Count of students in Class A with arm span > 170: \(11\).
  • Proportion for Class A: \(P_A = \frac{11}{23} \approx 0.478\).
  • Class B:
  • Values > 170: 173, 180, 181, 182, 189, 192, 202.
  • Count of students in Class B with arm span > 170: \(7\).
  • Proportion for Class B: \(P_B = \frac{7}{21} = \frac{1}{3} \approx 0.333\).
  1. Compare proportions and conclude

Comparing the two proportions:

  • Class A: \(P_A = \frac{11}{23} \approx 0.478\)
  • Class B: \(P_B = \frac{1}{3} \approx 0.333\)

Since \(0.478 > 0.333\), the proportion of arm spans longer than 170 cm is higher in Class A than in Class B. Therefore, the second statement is correct.

Examples

This type of analysis is common in educational statistics, such as comparing the physical development of students across different classes or grades. By visualizing data with a stemplot, researchers or teachers can quickly assess trends and distributions, like comparing the proportions of students meeting specific health or growth benchmarks (e.g., height or arm span) without needing to perform complex calculations first. For instance, if you were analyzing the distribution of test scores in two classrooms to decide where to focus additional tutoring, you would follow this exact process to identify which group is falling behind.

Related Questions