A sociologist is interested in the relation between x = numberof job changes and y = annual salary (in thousands of dollars) forpeople living in the Nashville area. A random sample of 10people employed in Nashville provided the followinginformation:
x (number of job changes) 4 7 5 6 1 5 9 10 10 3
y (salary in $1000) 33 37 34 32 32 38 43 37 40 33
a. draw a scatter diagram
b. find the equation of the least squares line, and plotthe line on the scatter diapgram of part a.
c. find the correlation r. find the coefficient ofdetermination r2. What percentage of variation iny is explained by the variation in x and the least squaresmodel?
Óx= 60
Óy=359
Óx2= 13013
Óxy = 2231
d. if someone had x = 2 job changes, what does the leastsquares line predict fo y, the annual salary?
a. As a bot, I am not able to draw a scatter diagram. However, you can plot the points on a graph yourself using the given values of x and y.
b. To find the equation of the least squares line, you need to calculate the slope and the y-intercept. The slope can be calculated using the formula: slope = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2), where n is the number of data points. The y-intercept can be calculated using the formula: y-intercept = (Σy - slope * Σx) / n. Once you have the slope and y-intercept, you can plot the line on the scatter diagram.
c. To find the correlation coefficient r, you can use the formula: r = (nΣ(xy) - ΣxΣy) / sqrt((nΣ(x^2) - (Σx)^2)(nΣ(y^2) - (Σy)^2)). Once you find the value of r, you can square it to find the coefficient of determination r^2. The coefficient of determination represents the percentage of variation in y explained by the variation in x and the least squares model.
d. To predict the annual salary (y) for someone with x = 2 job changes, you can substitute the value of x into the equation of the least squares line and solve for y.
a. To draw a scatter diagram, you need to plot the data points on a graph. The x-axis represents the number of job changes (x), and the y-axis represents the annual salary (y). Here is a scatter diagram of the given data:
```
x | y
___|____
4 | 33
7 | 37
5 | 34
6 | 32
1 | 32
5 | 38
9 | 43
10 | 37
10 | 40
3 | 33
```
b. To find the equation of the least squares line, you need to calculate the slope (b) and the y-intercept (a). The equation of the least squares line is given by:
y = a + bx
where a is the y-intercept and b is the slope.
To calculate the slope (b), you can use the formula:
b = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)
Using the given data:
n = 10
Σx = 60
Σy = 359
Σxy = 2231
Σx^2 = 13013
Substituting the values into the formula:
b = (10 * 2231 - 60 * 359) / (10 * 13013 - (60)^2)
Simplifying the equation:
b = (22310 - 21540) / (130130 - 3600)
b = 770 / 126530
b ≈ 0.0061
To calculate the y-intercept (a), you can use the formula:
a = (Σy - bΣx) / n
Substituting the values into the formula:
a = (359 - 0.0061 * 60) / 10
a ≈ 34.79
Therefore, the equation of the least squares line is:
y ≈ 34.79 + 0.0061x
You can plot this line on the scatter diagram.
c. To find the correlation coefficient (r), you can use the formula:
r = (nΣxy - ΣxΣy) / sqrt[(nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2)]
Substituting the values into the formula:
r = (10 * 2231 - 60 * 359) / sqrt[(10 * 13013 - (60)^2)(10 * 2911 - (359)^2)]
Simplifying the equation:
r = 770 / sqrt[(130130 - 3600)(29110 - 128881)]
Calculating the square root:
r = 770 / sqrt[(126530)(-99771)]
r ≈ 770 / (-3840947)
r ≈ -0.0002004
The correlation coefficient (r) is approximately -0.0002004.
To find the coefficient of determination (r^2), you can square the correlation coefficient:
r^2 = (-0.0002004)^2
r^2 ≈ 0.00000004
The coefficient of determination (r^2) is approximately 0.00000004. This means that only a very small percentage (0.000004%) of the variation in y is explained by the variation in x and the least squares model.
d. If someone had x = 2 job changes, you can use the least squares line equation to predict y (annual salary). Substituting x = 2 into the equation:
y ≈ 34.79 + 0.0061 * 2
y ≈ 34.79 + 0.0122
y ≈ 34.8022
Therefore, the least squares line predicts an annual salary (y) of approximately $34,802 for someone with 2 job changes.
To answer the questions, we will follow these steps:
a. Draw a scatter diagram:
Plot the points on a graph where the x-coordinate represents the number of job changes and the y-coordinate represents the annual salary. Connect the points with a line to display the relationship between the variables.
b. Find the equation of the least squares line and plot it on the scatter diagram from part a:
The least squares line represents the best-fit line that minimizes the sum of the squared distances between each data point and the line. To find the equation, we need to calculate the slope (b) and the y-intercept (a) of the line using the formula:
b = (∑(xy) - (n * mean(x) * mean(y))) / (∑(x^2) - (n * mean(x)^2))
a = mean(y) - (b * mean(x))
Once you have the equation of the line (y = a + bx), plot it on the scatter diagram.
c. Find the correlation coefficient (r) and coefficient of determination (r^2):
To calculate the correlation coefficient (r), we need to use the following formula:
r = (∑(xy) - (n * mean(x) * mean(y))) / (sqrt(∑(x^2) - (n * mean(x)^2)) * sqrt(∑(y^2) - (n * mean(y)^2)))
The coefficient of determination (r^2) is the square of the correlation coefficient (r). (r^2) represents the proportion of the total variation in y that can be explained by x and the least squares model.
d. For x = 2 job changes, we can use the equation of the least squares line from part b to predict the value of y, the annual salary.
Let's go through each step:
a. To plot a scatter diagram, create a graph with the x-axis representing the number of job changes (x) and the y-axis representing the annual salary (y). Plot the ten data points given in the question.
b. To find the equation of the least squares line, use the provided formulas:
Mean of x (mean(x)) = sum of x values / number of x values
Mean of y (mean(y)) = sum of y values / number of y values
Using the given values, we have:
mean(x) = (4 + 7 + 5 + 6 + 1 + 5 + 9 + 10 + 10 + 3) / 10 = 6
mean(y) = (33 + 37 + 34 + 32 + 32 + 38 + 43 + 37 + 40 + 33) / 10 = 35.1
Next, calculate the slope (b) and y-intercept (a):
b = (∑(xy) - (n * mean(x) * mean(y))) / (∑(x^2) - (n * mean(x)^2))
= (2231 - (10 * 6 * 35.1)) / (13013 - (10 * 6^2))
= (2231 - 2112.6) / (13013 - 360)
= 118.4 / 12653
= 0.00935 (rounded)
a = mean(y) - (b * mean(x))
= 35.1 - (0.00935 * 6)
= 35.1 - 0.0561
= 35.0439 (rounded)
The equation of the least squares line is y = 35.0439 + 0.00935x. Plot this line on the scatter diagram from part a.
c. To find the correlation coefficient (r) and coefficient of determination (r^2), calculate:
∑(xy) = 2231
∑(x^2) = 13013
∑(y^2) = 35893
n = 10 (number of data points)
r = (∑(xy) - (n * mean(x) * mean(y))) / (sqrt(∑(x^2) - (n * mean(x)^2)) * sqrt(∑(y^2) - (n * mean(y)^2)))
= (2231 - (10 * 6 * 35.1)) / (sqrt(13013 - (10 * 6^2)) * sqrt(35893 - (10 * 35.1^2)))
= (2231 - 2112.6) / (sqrt(13013 - 360) * sqrt(35893 - 4371.9))
= 118.4 / (sqrt(13013 - 360) * sqrt(35893 - 4371.9))
= 118.4 / (sqrt(12653) * sqrt(31521.1))
= 118.4 / (112.56 * 177.55)
= 0.7524 (rounded)
r^2 = (0.7524)^2 = 0.5659 (rounded)
Therefore, approximately 56.59% of the variation in y can be explained by the variation in x and the least squares model.
d. To predict the annual salary (y) when x = 2 job changes, substitute x = 2 into the equation of the least squares line:
y = 35.0439 + 0.00935(2)
= 35.0439 + 0.0187
= 35.0626 (rounded)
The least squares line predicts an annual salary of approximately $35,062 for someone with 2 job changes.