Answer to Question #328076 in Statistics and Probability for robot

Question

In 2016, around 12 Lakh students gave the CAT to bag an admission in India’s best business schools. In the math section of CAT, the marks follow a normal distribution with mean 520 and standard deviation 115.

A. What CAT score will be 1.5 standard deviations above the mean?

b. In 2022, a significant number of students also gave GMAT. Some colleges in India also accept GMAT scores as an alternative to CAT. The math section of GMAT is normally distributed with mean 21 and standard deviation 5.3. The math section is CAT-2022 also follows a normal distribution with mean 514 and standard deviation of 117. Raj took the CAT and scored 700 in the math section. Meanwhile, his friend Rahul scored 30 in the GMAT math section. Who do you think performed better? 

Answer 1

$a:\mu +1.5\sigma =520+1.5\cdot 115=692.5\\b: \\X\sim N\left( 514,117 \right) -Raj\\P\left( X\leqslant 700 \right) =P\left( \frac{X-514}{117}\leqslant \frac{700-514}{117} \right) =P\left( Z\leqslant 1.58974 \right) =\\=\varPhi \left( 1.58974 \right) \\Y\sim N\left( 21,5.3 \right) -Rahul\\P\left( Y\leqslant 30 \right) =P\left( \frac{Y-21}{5.3}\leqslant \frac{30-21}{5.3} \right) =P\left( Z\leqslant 1.69811 \right) =\\=\varPhi \left( 1.69811 \right) \\Since\,\,\varPhi \,\,is\,\,incre\mathrm{a}\sin g, \varPhi \left( 1.58974 \right) \leqslant \varPhi \left( 1.69811 \right) , which\,\,means\,\,Rahul\,\,is\,\,in\,\,higher\,\,percentile\,\,than\,\,Raj.\\Rahul\,\,performed\,\,better.$