Problem

The average minimum monthly temperature, $x$ degrees Fahrenheit $(^\circ\text{F})$, and the average maximum monthly temperature, $y$ degrees Fahrenheit $(^\circ\text{F})$, in Kolkata were recorded for $12$ months.

Some of the summary statistics are given below.

$$\sum x=862,\quad \sum x^2=62802,\quad S_{yy}=413.67,\quad S_{xy}=512.67,\quad n=12.$$

(a) (i) Calculate the mean of the $12$ values of the average minimum monthly temperature.

(ii) Show that the standard deviation of the $12$ values of the average minimum monthly temperature is $8.57^\circ\text{F}$ to $3$ significant figures.

(b) Calculate the product moment correlation coefficient between $x$ and $y$.

For comparative purposes with a UK city, it was necessary to convert the temperatures from degrees Fahrenheit $(^\circ\text{F})$ to degrees Celsius $(^\circ\text{C})$. The formula used was

$$c=\frac59(f-32)$$

where $f$ is the temperature in $^\circ\text{F}$ and $c$ is the temperature in $^\circ\text{C}$.

(c) Use this formula and the values from part (a) to calculate, in $^\circ\text{C}$, the mean and the standard deviation of the $12$ values of the average minimum monthly temperature in Kolkata. Give your answers to $3$ significant figures.

Given that

• $u$ is the equivalent temperature in $^\circ\text{C}$ of $x$
• $v$ is the equivalent temperature in $^\circ\text{C}$ of $y$

(d) state, giving a reason, the product moment correlation coefficient between $u$ and $v$.