A sociologist is interested in the relations between x=number of job changes and y=annual salary (in thousands of dollars) for people living in the Nashville area. A random sample of 10 people employed in Nashville provided the following information:

X(Number of job changes)4 7 5 6 1 5 9 10 10 3
Y (salary in $ 1,000) 33 37 34 32 32 38 43 37 40 33

What is your question?

To analyze the relations between the number of job changes and annual salary for people living in the Nashville area, we can calculate the correlation coefficient. The correlation coefficient measures the strength and direction of the linear relationship between two variables.

To calculate the correlation coefficient, you can follow these steps:

1. Create a table to organize the data. Label one column as X for the number of job changes and another column as Y for the corresponding annual salary.

X (Number of job changes) | Y (Salary in $1,000)
-------------------------|----------------------
4 | 33
7 | 37
5 | 34
6 | 32
1 | 32
5 | 38
9 | 43
10 | 37
10 | 40
3 | 33

2. Calculate the mean (average) for both X and Y. To find the mean, add up all the values in each column and divide by the total number of values.

Mean of X: (4 + 7 + 5 + 6 + 1 + 5 + 9 + 10 + 10 + 3) / 10 = 6
Mean of Y: (33 + 37 + 34 + 32 + 32 + 38 + 43 + 37 + 40 + 33) / 10 = 36.1

3. Calculate the difference between each value of X and the mean of X, and the difference between each value of Y and the mean of Y.

X - mean of X: (4 - 6) = -2
(7 - 6) = 1
(5 - 6) = -1
(6 - 6) = 0
(1 - 6) = -5
(5 - 6) = -1
(9 - 6) = 3
(10 - 6) = 4
(10 - 6) = 4
(3 - 6) = -3

Y - mean of Y: (33 - 36.1) = -3.1
(37 - 36.1) = 0.9
(34 - 36.1) = -2.1
(32 - 36.1) = -4.1
(32 - 36.1) = -4.1
(38 - 36.1) = 1.9
(43 - 36.1) = 6.9
(37 - 36.1) = 0.9
(40 - 36.1) = 3.9
(33 - 36.1) = -3.1

4. Multiply the differences obtained in step 3 (X - mean of X and Y - mean of Y) for each data point and find the sum of these products.

Sum of (X - mean of X) * (Y - mean of Y) = (-2 * -3.1) + (1 * 0.9) + (-1 * -2.1) + (0 * -4.1) + (-5 * -4.1) + (-1 * 1.9) + (3 * 6.9) + (4 * 0.9) + (4 * 3.9) + (-3 * -3.1) = 50.1

5. Calculate the standard deviation of X and Y. To find the standard deviation, follow these steps:

5.1 Calculate the squared difference between each value of X and the mean of X, and between each value of Y and the mean of Y.

(X - mean of X)^2: (-2)^2 = 4
(1)^2 = 1
(-1)^2 = 1
(0)^2 = 0
(-5)^2 = 25
(-1)^2 = 1
(3)^2 = 9
(4)^2 = 16
(4)^2 = 16
(-3)^2 = 9

(Y - mean of Y)^2: (-3.1)^2 = 9.61
(0.9)^2 = 0.81
(-2.1)^2 = 4.41
(-4.1)^2 = 16.81
(-4.1)^2 = 16.81
(1.9)^2 = 3.61
(6.9)^2 = 47.61
(0.9)^2 = 0.81
(3.9)^2 = 15.21
(-3.1)^2 = 9.61

5.2 Calculate the sum of the squared differences obtained in step 5.1.

Sum of (X - mean of X)^2 = 4 + 1 + 1 + 0 + 25 + 1 + 9 + 16 + 16 + 9 = 82
Sum of (Y - mean of Y)^2 = 9.61 + 0.81 + 4.41 + 16.81 + 16.81 + 3.61 + 47.61 + 0.81 + 15.21 + 9.61 = 124

5.3 Divide the sums obtained in step 5.2 by the total number of data points minus 1 (n - 1).

Standard deviation of X: sqrt(82 / (10 - 1)) ≈ 3.16
Standard deviation of Y: sqrt(124 / (10 - 1)) ≈ 3.97

6. Calculate the correlation coefficient using the formula:

correlation coefficient = (sum of (X - mean of X) * (Y - mean of Y)) / (n * standard deviation of X * standard deviation of Y)

correlation coefficient = 50.1 / (10 * 3.16 * 3.97) ≈ 0.399

The correlation coefficient for the given data is approximately 0.399. This indicates a moderate positive linear relationship between the number of job changes and the annual salary.