Question

The following problems pertain to the following information.

In a random sample of sixteen different industries the percentage employment x in STEM (science, technology, engineering, mathematics) and mean annual compensation y (in thousands of dollars) were recorded, with the following results. The scatter diagram showed a linear trend.

n=16   1≤x≤34   36.7≤y≤79.6     ∑x=150.1     ∑y=915

SS_xx = 1303.9         SS_xy = 1503.0         SS_yy = 2660.2

1. Find the linear correlation coefficient for percent employment in STEM and annual compensation.

2. Find the equation of the regression line for predicting y from x.

3. Predict the mean annual compensation in industries in which 30 percent of the workforce are in a STEM field.

4. Test whether the percentage employment x in STEM (science, technology, engineering, mathematics) is useful to predict the mean annual compensation y (in thousands of dollars).

Answer 1

Ʃx = 150.1

Ʃy = 915

Sample size, n = 16

x̅ = Ʃx/n = 150.1/16 = 9.38125

y̅ = Ʃy/n = 915/16 = 57.1875

SSxx = 1303.9

SSyy = 2660.2

SSxy = 1503

Slope, b = SSxy/SSxx = 1503/1303.9 = 1.1526958

y-intercept, a = y̅ -b* x̅ = 57.1875 - (1.1527)*9.38125 = 46.373773

a)

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 1503/√(1303.9*2660.2) = 0.8070

b)

Slope, b = SSxy/SSxx = 1503/1303.9 = 1.1526958

y-intercept, a = y̅ -b* x̅ = 57.1875 - (1.1527)*9.38125 = 46.373773

Regression equation :

ŷ = 46.3738 + (1.1527) x

c)

Predicted value of y at x = 30

ŷ = 46.3738 + (1.1527) * 30 = 80.9546

d)

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 2660.2 - (1503)²/1303.9 = 927.6982744

Standard error, se = √(SSE/(n-2)) = √(927.69827/(16-2)) = 8.14028

Null and alternative hypothesis:

Ho: β₁ = 0

Ha: β₁ ≠ 0

Test statistic:

t = b/(se/√SSxx) = 5.1133

df = n-2 = 14

p-value = T.DIST.2T(ABS(5.1133), 14) = 0.0002

Conclusion:

p-value < α Reject the null hypothesis.

There is enough evidence to conclude that the percentage employment x in STEM (science, technology, engineering, mathematics) is useful to predict the mean annual compensation y (in thousands of dollars).