A researcher is investigating the relationship between the distance from a sound source, x x\,x metres, and the measured intensity of the sound, y y\,y units.
The researcher measures 15 different distances and finds the product moment correlation coefficient between distance and sound intensity to be -0.48.
Using a 5% level of significance, test whether or not there is evidence of a negative correlation between distance and sound intensity. State your hypotheses clearly.
The researcher codes the data using m=log10xm = \log_{10}xm=log10x and n=log10yn = \log_{10}yn=log10y and obtains the model n=3.62−1.95mn = 3.62 - 1.95mn=3.62−1.95m.
The product moment correlation coefficient for the coded data is -0.98.
Explain why this supports the researcher's choice of a non-linear model.
Show that the relationship between x x\,x and y y\,y can be written in the form y=axby = ax^by=axb where a a\,a and b b\,b are constants to be found.