Question

Based on the two data sets given below, which of the following statements are true? Select TWO that apply. Data set A: 5, 8, 12, 15, 18, 20, 22, 25, 28, 30 Data set B: 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 Responses Data set A has a smaller mean than data set B. Data set A has a smaller mean than data set B. Data set A has a smaller standard deviation than data set B. Data set A has a smaller standard deviation than data set B. Data set A has a larger mean than data set B. Data set A has a larger mean than data set B. Data set A has a larger standard deviation than data set B. Data set A has a larger standard deviation than data set B.

Answer 1

To evaluate the statements, we first need to calculate the means and standard deviations of Data Set A and Data Set B.

**Data Set A:** 5, 8, 12, 15, 18, 20, 22, 25, 28, 30
**Data Set B:** 10, 12, 14, 16, 18, 20, 22, 24, 26, 28

### Calculating the Mean

**Mean of Data Set A:**
\[ \text{Mean A} = \frac{5 + 8 + 12 + 15 + 18 + 20 + 22 + 25 + 28 + 30}{10} = \frac{ 5 + 8 + 12 + 15 + 18 + 20 + 22 + 25 + 28 + 30 }{10} = \frac{ 158 }{10} = 15.8 \]

**Mean of Data Set B:**
\[ \text{Mean B} = \frac{10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28}{10} = \frac{ 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 }{10} = \frac{ 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 }{10} = \frac{ 20 + 22 + 24 + 26 + 28 }{10} = 19 \]

### Calculating Standard Deviation

The standard deviation formula is as follows:
\[ \sigma = \sqrt{\frac{1}{N} \sum (x_i - \mu)^2} \]

**For Data Set A:**
- Mean \( \mu_A = 15.8 \)
- Calculate variance:
\[
\begin{align*}
& (5 - 15.8)^2 = 116.64 \\
& (8 - 15.8)^2 = 60.84 \\
& (12 - 15.8)^2 = 14.44 \\
& (15 - 15.8)^2 = 0.64 \\
& (18 - 15.8)^2 = 4.84 \\
& (20 - 15.8)^2 = 17.64 \\
& (22 - 15.8)^2 = 38.44 \\
& (25 - 15.8)^2 = 83.64 \\
& (28 - 15.8)^2 = 148.84 \\
& (30 - 15.8)^2 = 202.44 \\
\end{align*}
\]

Sum of squared deviations:
\[
116.64 + 60.84 + 14.44 + 0.64 + 4.84 + 17.64 + 38.44 + 83.64 + 148.84 + 202.44 = 679.6
\]

Variance:
\[
\frac{679.6}{10} = 67.96
\]
Standard deviation:
\[
\sigma_A = \sqrt{67.96} \approx 8.23
\]

**For Data Set B:**
- Mean \( \mu_B = 19 \)
- Calculate variance:
\[
\begin{align*}
& (10 - 19)^2 = 81 \\
& (12 - 19)^2 = 49 \\
& (14 - 19)^2 = 25 \\
& (16 - 19)^2 = 9 \\
& (18 - 19)^2 = 1 \\
& (20 - 19)^2 = 1 \\
& (22 - 19)^2 = 9 \\
& (24 - 19)^2 = 25 \\
& (26 - 19)^2 = 49 \\
& (28 - 19)^2 = 81 \\
\end{align*}
\]

Sum of squared deviations:
\[
81 + 49 + 25 + 9 + 1 + 1 + 9 + 25 + 49 + 81 = 330
\]

Variance:
\[
\frac{330}{10} = 33
\]

Standard deviation:
\[
\sigma_B = \sqrt{33} \approx 5.74
\]

### Conclusion:
- Mean of Data Set A: \( 15.8 \)
- Mean of Data Set B: \( 19 \)
- So, **Data Set A has a smaller mean than Data Set B.** (This statement is true.)

- Standard deviation of Data Set A: \( \approx 8.23 \)
- Standard deviation of Data Set B: \( \approx 5.74 \)
- So, **Data Set A has a larger standard deviation than Data Set B.** (This statement is also true.)

### True Statements:
1. Data set A has a smaller mean than data set B.
2. Data set A has a larger standard deviation than data set B.