Answer to Question #130537 in Statistics and Probability for Samuel kassapa

Question

Temperature and air pollution are known to be correlated. We collect data from two laboratories, in Accra and Tema. Accra makes their measurements of temperature in Fahrenheit, and Tema in degrees centigrade. Accra measures pollution in particles per cubic yard of air; Tema uses cubic meters. Both report a correlation of exactly 0.58 between temperature and pollution. What conclusion can you draw from the results as to which one has a higher correlation?

A politician who is running for the office of governor of a state with 7 million registered voters commissions a survey. In the survey, 55% of the 10,000 registered voters interviewed say they plan to vote for her. What is the population of interest?

Answer 1

Answer 1)

The formula for Karl Pearson's coefficient of correlation is given as under-

$r = \frac{\sum(X-\bar{X})*(Y-\bar{Y})}{\sqrt{(\sum(X-\bar{X})^2)}\ *\ \sqrt{\sum(Y-\bar{Y})^2)}}$

In the given scenario we consider that,

X -> temperature data and

Y -> pollution data

$r = \frac{(units\ of\ measurement\ of\ temperature)*(units\ of\ measurement\ of\ pollution)}{\sqrt{(units\ of\ measurement\ of\ temperature)^2}*\sqrt{(units\ of\ measurement\ of\ pollution)^2}}$

$r = \frac{(units\ of\ measurement\ of\ temperature)*(units\ of\ measurement\ of\ pollution)}{(units\ of\ measurement\ of\ temperature)*(units\ of\ measurement\ of\ pollution)}$

But as we can see, if we substitute the values of X & Y in the above formula the units of measurements gets cancelled from the numerator and denominator and what remains is just the numerical value for 'r'.

Which means that coefficient of correlation 'r' is un-affected by the change in units of measurement or change of origin.

Hence we can conclude that the data from both the laboratories Accra and Tema has the same value of coefficient of correlation r = 0.58 and has exactly the same relation between temperature and pollution.

Answer 2)

A "population of interest" is defined as the population/group from which a researcher tries to draw conclusions.

In our case the population of interest for a politician who is running for the office of governor of a state is 7 million = 70,00,000