Problem

Stuart is investigating the relationship between Gross Domestic Product (GDP) and the size of the population for a particular country. He takes a random sample of $9$ years and records the size of the population, $t$ millions, and the GDP, $g$ billion dollars for each of these years.

The data are summarised as

$$n=9\qquad \sum t=7.87\qquad \sum g=144.84\qquad \sum g^2=3624.41$$ $$S_{tt}=1.29\qquad S_{tg}=40.25$$

(a) Calculate the product moment correlation coefficient between $t$ and $g$

(b) Give an interpretation of your product moment correlation coefficient.

(c) Find the equation of the least squares regression line of $g$ on $t$ in the form $g=a+bt$

(d) Give an interpretation of the value of $b$ in your regression line.

(e) (i) Use the regression line from part (c) to estimate the GDP, in billions of dollars, for a population of $7000000$

(ii) Comment on the reliability of your answer in part (i). Give a reason, in context, for your answer.

Using the regression line from part (c), Stuart estimates that for a population increase of $x$ million there will be an increase of $0.1$ billion dollars in GDP.

(f) Find the value of $x$