Question

(a) Calculate the value of 'a' for the dataset: a, 7.1, 6.8, 5.6, 3.2, 2.5, 2.5, 0.3, -1.2, -1.4, given that the sample variance is 11.361.

(b) Suppose x1, x2, ..., x8 is another set of 8 measurements with a sample mean of 2.4 and a sample variance of 7.52. Find the mean, variance, and standard deviation of the combined set of measurements when these two datasets are combined.

Answer 1

To calculate the value of 'a' for the dataset, we use the formula for sample variance. To find the mean, variance, and standard deviation of the combined set of measurements, we use formulas for mean and variance of combined datasets.

Step-by-step explanation:

To calculate the value of 'a' for the given dataset, we can use the formula for sample variance. The formula is: variance = sum((x - mean)^2) / (n - 1). Given that the sample variance is 11.361, we can set up the equation: 11.361 = (7.1 - a)^2 + (6.8 - a)^2 + ... + (-1.4 - a)^2. By solving this equation, we can find the value of 'a'.

To find the mean, variance, and standard deviation of the combined set of measurements when two datasets are combined, we can use the formulas for mean and variance of combined datasets. The mean of the combined set is the average of the individual sample means. The variance of the combined set is the sum of the individual variances plus the cross-product term. The standard deviation of the combined set is the square root of the variance.