Question

alannah and chloe are comparing the number of texts they each send per week. they each decide to look over their text logs and count their weekly sent texts over the past year. these data sets are then compiled into the following box plots:

box plot image: \texts sent throughout year\ with alannah and chloes box plots, x - axis \weekly sent messages\ from 10 to 80

based on these box plots, choose all of the following statements that are true.

show your work here

the range in the number of texts is equal for both alannah and chloe.
the range in the number of texts is larger for chloe than for alannah.
the maximum number of texts for alannah is 5 larger than the maximum number of texts for chloe.
the data for both alannah and chloe have an equal interquartile range (iqr).
the maximum number of texts for chloe is 10 larger than the maximum number of texts for alannah.

Explanation:

To solve this, we analyze the box - plots by recalling the definitions of range (maximum - minimum) and interquartile range (IQR = Q3 - Q1).

Step 1: Analyze the range

• For Alannah: Let's assume from the box - plot that the minimum value is \(m_A\) and the maximum value is \(M_A\). From the plot, we can see that the whiskers for Alannah seem to go from, say, \(20\) to \(70\) (approximate values from the axis). So the range for Alannah, \(R_A=M_A - m_A=70 - 20 = 50\).
• For Chloe: Let's assume the minimum value is \(m_C\) and the maximum value is \(M_C\). From the plot, the whiskers for Chloe seem to go from \(10\) to \(80\) (approximate values from the axis). So the range for Chloe, \(R_C = M_C - m_C=80 - 10 = 70\).
• Now, let's check the statements about the range:
• "The range in the number of texts is equal for both Alannah and Chloe." Since \(R_A = 50\) and \(R_C=70\), this statement is false.
• "The range in the number of texts is larger for Chloe than for Alannah." Since \(70>50\), this statement is true.
• "The maximum number of texts for Alannah is 5 larger than the maximum number of texts for Chloe." From the plot, Alannah's maximum is around \(70\) and Chloe's maximum is around \(80\). So Alannah's maximum is \(10\) less than Chloe's, not \(5\) larger. This statement is false.
• "The maximum number of texts for Chloe is 10 larger than the maximum number of texts for Alannah." Since \(80 - 70=10\), this statement is true.

Step 2: Analyze the Inter - Quartile Range (IQR)

• The IQR is calculated as \(IQR = Q_3-Q_1\), where \(Q_1\) is the first quartile and \(Q_3\) is the third quartile.
• Looking at the box - plots, the length of the box (which represents the inter - quartile range) for Alannah and Chloe:
• For Alannah, the box seems to span from \(40\) to \(60\) (approximate values from the axis), so \(IQR_A=60 - 40 = 20\).
• For Chloe, the box seems to span from \(20\) to \(70\) (approximate values from the axis), so \(IQR_C=70 - 20 = 50\).
• "The data for both Alannah and Chloe have an equal interquartile range (IQR)." Since \(20

eq50\), this statement is false.

Answer:

• The range in the number of texts is larger for Chloe than for Alannah.
• The maximum number of texts for Chloe is 10 larger than the maximum number of texts for Alannah.