Question

grace claims that the interquartile range of a data set is never equal to its range. which type of set can be used to prove or disprove her claim?
a set consisting of identical elements will prove her claim.
a set consisting of identical elements will disprove her claim.
a set consisting of different elements will disprove her claim.
a set consisting of different elements will prove her claim.

Explanation:

Let's consider a set with identical elements. For example, the set \(\{5,5,5,5\}\).

• Range: The range is calculated as \(max - min\). In this set, \(max = 5\) and \(min = 5\), so the range is \(5 - 5=0\).
• Inter - quartile range (IQR): First, we find the first quartile \(Q_1\) and the third quartile \(Q_3\). For the set \(\{5,5,5,5\}\), the median of the lower half (for \(Q_1\)) and the median of the upper half (for \(Q_3\)) are both \(5\). So, \(IQR=Q_3 - Q_1=5 - 5 = 0\).

If all elements are the same:

• The range (\(R=x_{max}-x_{min}\)) is \(0\) since \(x_{max}=x_{min}\).
• The inter - quartile range (\(IQR = Q_3 - Q_1\)) is \(0\) because \(Q_3 = Q_1\) (all values are equal).

For a set with different elements, say \(\{1,2,3,4\}\):

• Range \(R=4 - 1=3\).
• \(Q_1 = 1.5\), \(Q_3=3.5\), \(IQR=3.5 - 1.5 = 2

eq3\).

Answer:

A set consisting of identical elements will disprove her claim.