To solve this problem, let’s break down the information and use properties of variance.

Problem Statement

1. We are given a set of dataTo solve this problem, let’s break down the information and use properties of variance.

Problem Statement

1. We are given a set of $3, 5, 6, 7,$ and $9$.
2. The variance of this data set is $4$.
3. We are asked to find the variance of a new set of dataTo solve this problem, let’s break down the information and use properties of variance.

Problem Statement

1. We are given a set of $3, 5, 6, 7,$ and $9$.
2. The variance of this data set is $4$.
3. We are asked to find the variance of a new set of $6, 10, 12, 14,$ and $18$.

Analysis

Notice that each element in the second set is obtained by multiplying each element in the first set by $2$: $6 = 2 \times 3, \quad 10 = 2 \times 5, \quad 12 = 2 \times 6, \quad 14 = 2 \times 7, \quad 18 = 2 \times 9$

Variance Scaling Property

If we multiply each element in a data set by a constant $k$, the variance of the new data set becomes $k^2$ times the variance of the original data set.

Here:

• The original variance $= 4$.
• The scaling factor $k = 2$, so the new variance will be: $\text{New Variance} = k^2 \times (\text{Original Variance}) = 2^2 \times 4 = 4 \times 4 = 16$

Conclusion

The variance of the new data set $6, 10, 12, 14,$ and $18$ is $16$.

Answer

D. 16

Would you like further details on this solution, or have any questions?

Related Questions

1. How would the variance change if each element of a data set is increased by a constant instead of multiplied?
2. What is the relationship between standard deviation and variance when scaling a data set?
3. How can we find the mean of the new data set based on the original data set?
4. If the variance of a data set is zero, what does that indicate about the data points?
5. How do we calculate the variance of a data set from scratch if not given?

Tip

Always remember that variance scales with the square of the multiplier when each data point is scaled by a constant.