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?
(1 point)
Responses
The mean time taken by Group A is higher than that of Group B.
The ranges of the time taken by Group A and Group B are the same.
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 median time members of Group A take to thread a needle is higher than that of members of Group B.
The false statement is: The ranges of the time taken by Group A and Group B are the same.
To compare the measures of center and variability of the two data sets, let's calculate the mean, range, standard deviation, and median for both Group A and Group B.
Group A: 56, 14, 26, 28, 29, 45, 32, 80, 3, 10.
Mean (average) of Group A:
(56 + 14 + 26 + 28 + 29 + 45 + 32 + 80 + 3 + 10) / 10 = 323 / 10 = 32.3
Range of Group A:
The range is calculated by subtracting the minimum value from the maximum value.
Range = Maximum value - Minimum value = 80 - 3 = 77
Standard Deviation of Group A:
To calculate the standard deviation, we need to find the variance first and then take the square root of it. Variance measures the variability in a dataset.
1. Calculate the variance:
- Subtract the mean from each data point, square the differences, and sum them.
(56 - 32.3)^2 + (14 - 32.3)^2 + (26 - 32.3)^2 + (28 - 32.3)^2 + (29 - 32.3)^2 + (45 - 32.3)^2 + (32 - 32.3)^2 + (80 - 32.3)^2 + (3 - 32.3)^2 + (10 - 32.3)^2 = 1784.14
2. Divide the variance by the sample size (n) - 1.
1784.14 / (10 - 1) = 198.24
3. Take the square root to find the standard deviation.
√(198.24) ≈ 14.07
Median of Group A:
Arrange the numbers in ascending order: 3, 10, 14, 26, 28, 29, 32, 45, 56, 80.
Median is the middle value: 29.
Group B: 27, 24, 18, 19, 5, 16, 22, 10, 6, 20.
Mean (average) of Group B:
(27 + 24 + 18 + 19 + 5 + 16 + 22 + 10 + 6 + 20) / 10 = 167 / 10 = 16.7
Range of Group B:
Range = Maximum value - Minimum value = 27 - 5 = 22
Standard Deviation of Group B:
1. Calculate the variance:
(27 - 16.7)^2 + (24 - 16.7)^2 + (18 - 16.7)^2 + (19 - 16.7)^2 + (5 - 16.7)^2 + (16 - 16.7)^2 + (22 - 16.7)^2 + (10 - 16.7)^2 + (6 - 16.7)^2 + (20 - 16.7)^2 = 229.14
2. Divide the variance by the sample size (n) - 1.
229.14 / (10 - 1) = 28.64
3. Take the square root to find the standard deviation.
√(28.64) ≈ 5.35
Median of Group B:
Arrange the numbers in ascending order: 5, 6, 10, 16, 18, 19, 20, 22, 24, 27.
Median is the middle value: 18.
Based on these calculations, the false statement is:
The median time members of Group A take to thread a needle is higher than that of members of Group B.
The median time for Group A is 29, which is lower than the median time for Group B, which is 18.