Statistics Questions with Solutions

If ${\sum }_{i=1}^n{x}_i=80$ and ${\sum }_{i=1}^n{x}_i^2=400$, then a possible value of n among the following is:

Let $x_i$, $i=1,2,\dots ,n$ be n observations and $w_i=p{x}_i+k$, $i=1,2,\dots ,n$, where p and k are constants. If the mean of the $x_i$ 's is 48 and their standard deviation is 12, whereas the mean of the $w_i$ 's is 55 and their standard deviation is 15, then the values of p and k should be:

In a reality show, two judges independently provided marks based on the performance of the participants. The marks provided by the second judge are given by $Y=10.5+2X$, where $X$ is the marks provided by the first judge. If the variance of the marks provided by the second judge is 100, then the variance of the marks provided by the first judge is

It is given that the mean, median and mode of a data set are $1$, $3^x$ and $9^x$ respectively. The possible values of the mode are

Which one of the following is NOT a correct statement?