Question

6. 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

Sample Mean	x̄ x̄ = 4	ȳ ȳ = 5	z̄ z̄ = 5
Sample Variance	sx2sx2 = 4	sy2sy2 = 3	sz2sz2 = 9
Sample Standard Deviation	sxsx = 2	sysy = 1.732	szsz = 3

Table 2 shows the observations for x and y and their corresponding deviations from the sample means.

Table 2

xixi	yiyi	xixi – x̄ x̄	yiyi – ȳ ȳ
6	6	2	1
4	3	0	–2
2	6	–2	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

yiyi	zizi	yiyi – ȳ ȳ	zizi – z̄ z̄
6	2	1	–3
3	8	–2	3
6	5	1	0

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

xixi	zizi	xixi – x̄ x̄	zizi – z̄ z̄
6	2	2	–3
4	8	0	3
2	5	–2	0

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 linear relationship between x and y, a positive linear relationship between y and z, and no relationship between x and z.

The calculations show no linear relationship between x and y, a negative linear relationship between y and z, and a negative linear relationship between x and z.

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, a negative linear relationship between y and z, and a negative linear relationship between x and z.

Answer 1

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