Question

The mean and standard deviation of 100 observations are 40 and 5.1, respectively, By mistake one observation is taken as 50 instead of 40. If the correct mean and the correct standard deviation are \(\mu\) and \(\sigma\) respectively, then \(10(\mu+\sigma)\) is equal to

(1) 445

(2) 451

(3) 447

(4) 449 

Answer 1

Correct option is: (4) 449  

Actual means \(=\mu=\frac{100(40)-50+40}{100}\)

\(\mu=40-\frac{1}{10}=39.9\)

Incorrect variance

\((5.1)^{2}=\frac{\sum x_{i}^{2}}{100}-(\overline{\mathrm{x}})^{2} \)

\( \sum \mathrm{x}_{\mathrm{i}}^{2}=100 \times\left(40^{2}\right)+100(5.1)^{2} \)

\(\sum \mathrm{x}_{\mathrm{i}}^{2}=16 \times 10^{4}+(5.1)^{2} \times 100=162601 \)

\( \sigma^{2}=\frac{\sum \mathrm{x}_{\mathrm{i}}^{2}-50^{2}+40^{2}}{100}-(\mu)^{2} \)

\( \sigma^{2}=1617.01-(39.9)^{2}=25 \)

\(\sigma=5 \)

\( 10(\mu+\sigma)=10(39.9+5) \)

\( =10 \times 44.9=449\)