The accompanying summary quantities for x= particulate pollution and y = luminan
ID: 3363036 • Letter: T
Question
The accompanying summary quantities for x= particulate pollution and y = luminance were calculated from a representative sample of data that appeared in the article "Luminance and Polarization of the Sky Light at Seville (Spain) Measured in White Light" (Atmospheric Environment [1988]).
n= 15, x= 860 , y = 348, x2= 56,700, y2= 8,954, xy = 22,265
a. test to see whether there is a postive correlation between particulate pollution and luminance in the population from which the data were selected.
b. What proportion of observed variation in luminance can be attributed to the approxiamate linear relationship between luminance and particulate pollution?
Explanation / Answer
a.
Sxx = x2 - (x)2 / n = 56700 - 8602 / 15 = 7393.333
Syy = y2 - (y)2 / n = 8954 - 3482 / 15 = 880.4
Sxy = xy - (xy)/ n = 22265 - 860*348 / 15 = 2313
Correlation, r = Sxy / sqrt(Sxx * Syy) = 2313 / sqrt(7393.333 * 880.4) = 0.9066
Null Hypothesis H0: r = 0
Alternative Hypothesis H0: r > 0
The critical value of r for n = 15 and significance level of 0.05 is 0.441
As, the observed correlation (0.9066) is greater than the critical value (0.441), we reject the null hypothesis and conclude that there is a postive correlation between particulate pollution and luminance in the population from which the data were selected.
b.
Coefficient of determination is, r2 = 0.9066 * 0.9066 = 0.8219
So, 0.8219 (82.19%) of observed variation in luminance can be attributed to the approxiamate linear relationship between luminance and particulate pollution.
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