people at a library were asked how many states they had visited. below is a stemplot of all of the data…

people at a library were asked how many states they had visited. below is a stemplot of all of the data except for one value. which of the following statements is true? if the missing data value were 32, the iqr would change substantially. if the missing data value were 32, the range would change substantially. if the missing data value were 50, the iqr of the data would change substantially. if the missing data value were 50, the range of the data would change substantially. number of states visited 0 8 9 9 9 1 0 1 2 2 2 4 5 6 6 8 8 2 1 2 7 7 7 3 1 key: 2|7 = 27 states

people at a library were asked how many states they had visited. below is a stemplot of all of the data except for one value. which of the following statements is true? if the missing data value were 32, the iqr would change substantially. if the missing data value were 32, the range would change substantially. if the missing data value were 50, the iqr of the data would change substantially. if the missing data value were 50, the range of the data would change substantially. number of states visited 0 8 9 9 9 1 0 1 2 2 2 4 5 6 6 8 8 2 1 2 7 7 7 3 1 key: 2|7 = 27 states

Answer

Explanation:

Step1: Recall the definitions

  • Range is ( \text{Max}-\text{Min} ).
  • IQR is ( Q_{3}-Q_{1} ).

Step2: Analyze each option

  • Option 1:
    • Original data (without the missing value): ( n = 19 ) (count the number of data points in the stemplot).
    • ( Q_{1} ): For ( n = 19 ), ( Q_{1} ) is the ( \frac{n + 1}{4}=\frac{19+1}{4}=5^{th} ) value. The ordered data (from stemplot): ( 8,9,9,9,10,11,12,12,12,14,15,16,16,18,18,21,22,27,27,27,31 ). ( Q_{1}=12 ), ( Q_{3} ) is the ( \frac{3(n + 1)}{4}=\frac{3\times(19 + 1)}{4}=15^{th} ) value. ( Q_{3}=18 ).
    • If the missing value is ( 32 ), ( n = 20 ). ( Q_{1} ) is the ( \frac{n}{4}=5^{th} ) value (average of ( 5^{th} ) and ( 6^{th} ) for even ( n ) in some cases, but since the data is ordered ( 8,9,9,9,10,11,12,12,12,14,15,16,16,18,18,21,22,27,27,27,31,32 )), ( Q_{1}=\frac{12 + 12}{2}=12 ), ( Q_{3}=\frac{21+22}{2} = 21.5 ). The change in IQR (( 18-12 = 6 ) vs ( 21.5-12=9.5 )) is not substantial.
  • Option 2:
    • Original range (without the missing value): ( 31-8 = 23 ). If the missing value is ( 32 ), new range ( 32 - 8=24 ). The change is not substantial.
  • Option 3:
    • If the missing value is ( 50 ), ( Q_{1} ) (calculated as above for ( n = 20 )): ( Q_{1}=\frac{12 + 12}{2}=12 ), ( Q_{3}=\frac{27+27}{2}=27 ). The change in IQR (( 18-12 = 6 ) vs ( 27 - 12=15 )) is not the main point.
  • Option 4:
    • Original range (without the missing value): ( 31-8 = 23 ).
    • If the missing value is ( 50 ), new range ( 50-8 = 42 ). The change from ( 23 ) to ( 42 ) is substantial.

Answer:

If the missing data value were 50, the range of the data would change substantially.