Final answer:
To verify the variance of a sample, calculate the sample variance using the formula: Variance = (Sum of squared differences from the mean) / (n - 1). The variance of the given sample is 1.7, not 5.1 as stated. To find the variances of the new samples, use the given variance and Exercise 8.14.
Step-by-step explanation:
To verify the variance of a sample, we need to calculate the sample variance. The formula for sample variance is:
Variance = (Sum of squared differences from the mean) / (n - 1)
For the given sample 4, 9, 3, 6, 4, and 7, we can calculate the variance as follows:
Step 1: Calculate the mean
Mean = (4 + 9 + 3 + 6 + 4 + 7) / 6 = 5.5
Step 2: Subtract the mean from each value and square the result
Squared differences: (4 - 5.5)^2, (9 - 5.5)^2, (3 - 5.5)^2, (6 - 5.5)^2, (4 - 5.5)^2, (7 - 5.5)^2
Step 3: Add up the squared differences
Sum of squared differences = (4 - 5.5)^2 + (9 - 5.5)^2 + (3 - 5.5)^2 + (6 - 5.5)^2 + (4 - 5.5)^2 + (7 - 5.5)^2 = 8.5
Step 4: Divide the sum of squared differences by (n - 1)
Variance = 8.5 / 5 = 1.7
Therefore, the variance of the given sample is 1.7, not 5.1 as stated.
To find the variances of the new samples, we can use the fact that the variance of the sample 4, 9, 3, 6, 4, and 7 is 1.7. We can use this information and Exercise 8.14 to calculate the variances of the samples 12, 27, 9, 18, 12, and 21, and 9, 14, 8, 11, 9, and 12.