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 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 :

  • Total students: Class A (\(N_A=24\)), Class B (\(N_B=20\)).
  • Students with arm span > 170 cm: Class A (\(n_A=11\)), Class B (\(n_B=7\)).
  • Proportions: \(p_A = \frac{11}{24} \approx 0.458\) and \(p_B = \frac{7}{20} = 0.35\).
  • Since \(p_A > p_B\), the proportion is higher in Class A.
  • Final Answer: \(\boxed{\text{The proportion of arm spans that are longer than 170 cm is higher in Class A than in Class B.}}\)

Explanation

  1. Identify and list all data points from the stemplot

First, we identify the data for each class from the provided stemplot.
For Class A, the values are: 148, 153, 151, 159, 156, 155, 164, 162, 161, 169, 169, 165, 171, 170, 179, 179, 176, 175, 183, 182, 180, 186, 186, 190.
There are a total of \(24\) students in Class A.
For Class B, the values are: 153, 155, 156, 160, 162, 162, 162, 163, 163, 165, 166, 167, 170, 173, 180, 181, 182, 189, 192, 202.
There are a total of \(20\) students in Class B.

  1. Count the number of students with arm spans > 170 cm

Next, we count how many students in each class have an arm span strictly greater than \(170 \text{ cm}\).

  • For Class A, we look for values greater than \(170\). These are: \(171, 179, 179, 176, 175, 183, 182, 180, 186, 186, 190\). There are \(11\) such values.
  • For Class B, we look for values greater than \(170\). These are: \(173, 180, 181, 182, 189, 192, 202\). There are \(7\) such values.
  1. Calculate and compare the proportions

Now, we calculate the proportions for each class:

  • For Class A: \(p_A = \frac{11}{24} \approx 0.458\).
  • For Class B: \(p_B = \frac{7}{20} = 0.35\).

Comparing the two, we see that \(0.458 > 0.35\). Thus, the proportion of arm spans longer than \(170 \text{ cm}\) is higher in Class A than in Class B.

  1. Formulate the conclusion

Based on the calculations, we conclude that the proportion of students with arm spans longer than \(170 \text{ cm}\) is higher in Class A. Therefore, the second statement is correct.

Examples

Understanding how to compare data sets is a fundamental skill in statistics. For example, a sports coach might use this method to compare the physical characteristics of players from two different teams, helping them decide on training strategies tailored to each group's average size or specific physical attributes. By visualizing data through tools like stemplots, we can quickly extract key statistics to inform our decisions.

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