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Number of Pairs of Shoes Number of 6th Graders 2 4 3 3 4 1 5 1 6 3 7 7 8 2 10 1 Michael wants to know how many pairs of shoes 6th graders own. He surveys the 6th graders at his summer camp. His results are given in the table. After Michael creates the table, he realizes that he forgot to include himself. Michael owns 4 pairs of shoes. Determine the interquartile range (IQR) of the original dataset and the interquartile range of the new dataset with the data point for Michael’s 4 pairs of shoes added.

1 answer

Let's begin by calculating the interquartile range (IQR) of the original dataset.

Step 1: Organize the Original Dataset

The original dataset represents the number of pairs of shoes owned by 6th graders, based on the table provided:

  • 2 pairs: 4 students
  • 3 pairs: 3 students
  • 4 pairs: 1 student
  • 5 pairs: 1 student
  • 6 pairs: 3 students
  • 7 pairs: 7 students
  • 8 pairs: 2 students
  • 10 pairs: 1 student

To find the IQR, we first create a sorted list with the number of pairs represented in the dataset as follows:

  • 2, 2, 2, 2 (4 students with 2 pairs)
  • 3, 3, 3 (3 students with 3 pairs)
  • 4 (1 student with 4 pairs)
  • 5 (1 student with 5 pairs)
  • 6, 6, 6 (3 students with 6 pairs)
  • 7, 7, 7, 7, 7, 7, 7 (7 students with 7 pairs)
  • 8, 8 (2 students with 8 pairs)
  • 10 (1 student with 10 pairs)

Putting them together, we have:

\[ 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 10 \]

Step 2: Find the Quartiles for the Original Dataset

The dataset contains 22 values. To find Q1 and Q3 for the IQR calculation:

  • Position for Q1 (1st quartile): \(\frac{22 + 1}{4} = 5.75\). This means Q1 is the average of the 5th and 6th values.
  • Position for Q3 (3rd quartile): \(\frac{3 \times (22 + 1)}{4} = 17.25\). This means Q3 is the average of the 17th and 18th values.

Calculating Q1:

  • The 5th value is 3, and the 6th value is 3.
  • Q1 = \(\frac{3 + 3}{2} = 3\)

Calculating Q3:

  • The 17th value is 7, and the 18th value is 7.
  • Q3 = \(\frac{7 + 7}{2} = 7\)

Step 3: Calculate the IQR for the Original Dataset

\[ \text{IQR} = Q3 - Q1 = 7 - 3 = 4 \]

Step 4: Add Michael's Data Point

Michael owns 4 pairs of shoes. We will add this to the dataset:

  • Updated dataset: \[ 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 10 \]

This dataset now has 23 values.

Step 5: Find the Quartiles for the New Dataset

Finding Q1 and Q3 again:

  • Position for Q1: \(\frac{23 + 1}{4} = 6\). This refers to the 6th value.
  • Position for Q3: \(\frac{3 \times (23 + 1)}{4} = 17.25\). This refers to the average of the 17th and 18th values.

Calculating Q1:

  • The 6th value is 3.
  • Q1 = 3

Calculating Q3:

  • The 17th value is 7, and the 18th value is 7.
  • Q3 = \( \frac{7 + 7}{2} = 7 \)

Step 6: Calculate the IQR for the New Dataset

\[ \text{IQR} = Q3 - Q1 = 7 - 3 = 4 \]

Summary

  • The IQR of the original dataset is 4
  • The IQR of the new dataset (including Michael's data point) is also 4