As the price of a product increases, people are less likely to buy it. Greater the number of hours spent exercising, greater will be the reduction in percentage body fat. These two examples show a negative correlation.

Here, Y decreases with an increase in X. Hence the correlation is negative.

Correlation does not imply causation. Even if the independent variable and the dependent variable are interchanged, the correlation remains the same. Hence, the statement is false.

$\begin{array}{cccccccc}x& y& x-\overline{x}& (x-\overline{x}{)}^2& (y-\overline{y})& (y-\overline{y}{)}^2& (x-\overline{x})(y-\overline{y})& \\ 2& 4& -3& 9& -6& 36& 18& \\ 3& 6& -2& 4& -4& 16& 8& \\ 4& 9& -1& 1& -1& 1& 1& \\ 5& 10& 0& 0& 0& 0& 0& \\ 6& 11& 1& 1& 1& 1& 1& \\ 7& 13& 2& 4& 3& 9& 6& \\ 8& 17& 3& 9& 7& 49& 21& \\ & & & \sum =28& & \sum =112& \sum =55& \end{array}$

$\overline{x}=\frac{\sum x}{n}=\frac{35}{7}=5$
$\overline{y}=\frac{\sum y}{n}=\frac{70}{7}=10$

$r=\frac{{\sum }_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{\sum }_{i=1}^n(x_i-\overline{x}{)}^2}\sqrt{{\sum }_{i=1}^n(y_i-\overline{y}{)}^2}}=\frac{55}{\sqrt{28\times 112}}=\frac{55}{56}=0.982$

Draw a line parallel to the y-axis through x=10 to intersect the line of best fit. Through this point, draw a line parallel to the x-axis to intersect the y-axis. It can be seen that this line cuts y-axis near 30. Hence the approximate value of y corresponding to x=10 is 30.

$\begin{array}{ccccc}X& Y& X-\overline{X}& Y-\overline{Y}& (X-\overline{X})(Y-\overline{Y})\\ 7& 20& -4& -14& 56\\ 9& 25& -2& -9& 18\\ 11& 36& 0& 2& 0\\ 13& 41& 2& 7& 14\\ 15& 48& 4& 14& 56\\ & & & & \sum =144\end{array}$

$\overline{X}=\frac{\sum X}{N}=\frac{55}{5}=11$
$\overline{Y}=\frac{\sum Y}{N}=\frac{170}{5}=34$

$Cov(X,Y)=\frac{(X-\overline{X})(Y-\overline{Y})}{N}=\frac{144}{5}=28.8$

False. Spearman’s Rank coefficient helps us understand the correlation between the variables which can’t be very meaningful and predominantly subjective. It is generally used to find the correlation in case of qualitative variables. 

$\begin{array}{ccccc}{X}_i& Y_i& X_i^2& Y_i^2& X_i{Y}_i\\ 3& 75& 9& 5625& 225\\ 4& 85& 16& 7225& 340\\ 5& 80& 25& 6400& 400\\ 6& 82& 36& 6724& 492\\ 7& 88& 49& 7744& 616\\ \sum =25& \sum =410& \sum =135& \sum =33718& \sum =2073\end{array}$
$r=\frac{N{\sum }_{i=1}^n\left(X_i{Y}_i\right)–\left({\sum }_{i=1}^n{X}_i\right)\left({\sum }_{i=1}^n{Y}_i\right)}{\sqrt{N{\sum }_{i=1}^n{X}_i^2–({\sum }_{i=1}^n{X}_i{)}^2}\sqrt{N{\sum }_{i=1}^n{Y}_i^2–({\sum }_{i=1}^n{Y}_i{)}^2}}=\frac{115}{7.07\times 22.14}=0.735$

$\begin{array}{ccccc}{X}_i& Y_i& X_i^2& Y_i^2& X_i{Y}_i\\ 10& 120& 100& 14400& 1200\\ 20& 230& 400& 52900& 4600\\ 30& 315& 900& 99225& 9450\\ 40& 425& 1600& 180625& 17000\\ 50& 510& 2500& 260100& 25500\\ \sum =150& \sum =1600& \sum =5500& \sum =607250& \sum =57750\end{array}$
$r=\frac{N{\sum }_{i=1}^n\left(X_i{Y}_i\right)–\left({\sum }_{i=1}^n{X}_i\right)\left({\sum }_{i=1}^n{Y}_i\right)}{\sqrt{N{\sum }_{i=1}^n{X}_i^2–({\sum }_{i=1}^n{X}_i{)}^2}\sqrt{N{\sum }_{i=1}^n{Y}_i^2–({\sum }_{i=1}^n{Y}_i{)}^2}}=\frac{48750}{70.71\times 690.11}=0.999$