Question

Based on the two data sets given below, which of the following statements are true? Select two that apply.


Data Set 1: 5, 5, 6, 6, 7, 7
Data Set 2: 1, 3, 5, 7, 9, 11
(1 point)
Responses

Data Set 2 has a smaller standard deviation than Data Set 1.
Data Set 2 has a smaller standard deviation than Data Set 1.

Data Set 2 has the same mean as Data Set 1.
Data Set 2 has the same mean as Data Set 1.

Data Set 2 has a larger standard deviation than Data Set 1.
Data Set 2 has a larger standard deviation than Data Set 1.

Data Set 2 has a larger mean than Data Set 1.

Answer 1

To analyze the two data sets and determine which statements are true, we first need to calculate the means and standard deviations of both data sets.

**Data Set 1: 5, 5, 6, 6, 7, 7**
- Mean = (5 + 5 + 6 + 6 + 7 + 7) / 6 = 36 / 6 = 6
- To calculate the standard deviation, we first find the variance:
- Variance = [(5-6)² + (5-6)² + (6-6)² + (6-6)² + (7-6)² + (7-6)²] / 6
- Variance = [1 + 1 + 0 + 0 + 1 + 1] / 6 = 4 / 6 = 2/3
- Standard deviation = √(2/3) = √(0.6667) ≈ 0.8165

**Data Set 2: 1, 3, 5, 7, 9, 11**
- Mean = (1 + 3 + 5 + 7 + 9 + 11) / 6 = 36 / 6 = 6
- To calculate the standard deviation:
- Variance = [(1-6)² + (3-6)² + (5-6)² + (7-6)² + (9-6)² + (11-6)²] / 6
- Variance = [25 + 9 + 1 + 1 + 9 + 25] / 6 = 70 / 6 ≈ 11.6667
- Standard deviation = √(11.6667) ≈ 3.4142

Now we can evaluate the statements:

1. **Data Set 2 has a smaller standard deviation than Data Set 1.** - **False.** (Data Set 2 has a larger standard deviation).

2. **Data Set 2 has the same mean as Data Set 1.** - **True.** (Both data sets have a mean of 6).

3. **Data Set 2 has a larger standard deviation than Data Set 1.** - **True.** (Data Set 2 has a standard deviation of approximately 3.4142, which is larger than Data Set 1’s standard deviation of approximately 0.8165).

4. **Data Set 2 has a larger mean than Data Set 1.** - **False.** (Both have the same mean).

Based on this analysis, the two true statements are:

- Data Set 2 has the same mean as Data Set 1.
- Data Set 2 has a larger standard deviation than Data Set 1.