3. Measures of linear relationship - Covariance and coefficient of correlation Consider a data set consisting of observations for three variables: x, y, and z. Their sample means, variances, and standard deviations are shown in Table 1. Table 1 x = 3 y = 6 z = 2 Sample Mean Sample Variance Sample Standard Deviation Sy2 = 3 Sy2 = 3 Sy2 = 4 Sx = 1.732 Sy = 1.732 Sz = 2 Table 2 shows the observations for x and y and their corresponding deviations from the sample means. Table 2 Xi Xi - x yi - y 5 7 2 1 2 4 -1 -2 2 7 -1 1 The sample covariance between x and y is The sample correlation coefficient between x and y is Table 3 shows the observations for y and z and their corresponding deviations from the sample means. Table 3 yi Zi Yi - y Zi - Z 7 4 1 2 4 2 -2 0 7 0 1 -2 The sample covariance between y and z is The sample correlation coefficient between y and z is Table 4 shows the observations for x and z and their corresponding deviations from the sample means. Table 4 Xi Zi X - x Zi - Z 5 4 2 2. 2 2 -1 0 2 0 -1 -2 The sample covariance between x and z is The sample correlation coefficient between x and z is Select the best conclusion based on your calculations of the preceding correlation coefficients. The calculations show a negative relationship between x and y, a positive relationship between y and z, and a negative relationship between x and z. The calculations show a positive linear relationship between x and y, no linear relationship between y and z, and a positive linear relationship between x and z. O The calculations show a negative linear relationship between x and y, a positive linear relationship between y and z, and no relationship between x and z. O The calculations show a positive linear relationship between x and y, a negative linear relationship between y and z, and a negative linear relationship between x and z