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The data shows the number of seconds it took two randomly selected groups to thread a needle. Compare the measures of center and variability of these data sets.

Student Group A: 56, 14, 26, 28, 29, 45, 32, 80, 3, 10
Student Group B: 27, 24, 18, 19, 5, 16, 22, 10, 6, 20

Which of the following statements about these data sets is false?

Responses

The median time taken by members of Group A to thread a needle is higher than that of members of Group B.
The median time taken by members of Group A to thread a needle is higher than that of members of Group B.

The mean time taken by Group A is higher than that of Group B.
The mean time taken by Group A is higher than that of Group B.

The amount of variability of the time taken by Group A to thread a needle is higher than the amount of variability of the time taken by Group B.
The amount of variability of the time taken by Group A to thread a needle is higher than the amount of variability of the time taken by Group B.

The ranges of the time taken by Group A and Group B are the same.

1 answer

To compare the measures of center and variability of the two groups, we can calculate the mean, median, and range for both sets of data.

Group A: 56, 14, 26, 28, 29, 45, 32, 80, 3, 10

  1. Sort the data: 3, 10, 14, 26, 28, 29, 32, 45, 56, 80
  2. Median: Middle value (5th and 6th values averaged) = \( (28 + 29) / 2 = 28.5 \)
  3. Mean: \[ \text{Mean} = \frac{3 + 10 + 14 + 26 + 28 + 29 + 32 + 45 + 56 + 80}{10} = \frac{ 24 + 80 + 10 + 14 + 26 + 28 + 29 + 32 + 45 + 56}{10} = \frac{ 20 + 80 + 10 + 14 + 26 + 28 + 29 + 32 + 45 + 56 + 5 + 6 + 40 + 25 + 20}{10}= 30.3 \]
  4. Range: Maximum - Minimum = 80 - 3 = 77

Group B: 27, 24, 18, 19, 5, 16, 22, 10, 6, 20

  1. Sort the data: 5, 6, 10, 16, 18, 19, 20, 22, 24, 27
  2. Median: Middle value (5th and 6th values averaged) = \( (18 + 19) / 2 = 18.5 \)
  3. Mean: \[ \text{Mean} = \frac{5 + 6 + 10 + 16 + 18 + 19 + 20 + 22 + 24 + 27}{10} = \frac{ 27 + 12 + 10 + 8 + 3 + 21 + 24 + 25 + 20 + 11 + 16 + 19 + 15 + 29 + 24}{10} = 17.55 \]
  4. Range: Maximum - Minimum = 27 - 5 = 22

Summary:

  • Group A:

    • Median: 28.5
    • Mean: 30.3
    • Range: 77

  • Group B:

    • Median: 18.5
    • Mean: 17.55
    • Range: 22

Comparing the Groups:

  1. The median time for Group A (28.5) is higher than that for Group B (18.5).
  2. The mean time for Group A (30.3) is higher than that for Group B (17.55).
  3. The variability of Group A (range of 77) is higher than that of Group B (range of 22).

False Statement:

Based on this comparison, the statement "The ranges of the time taken by Group A and Group B are the same." is false, as the range for Group A is 77, and for Group B is 22, hence they are not the same.