Answer to Question #320897 in Statistics and Probability for Neil

Question

the mean score and the standard score in the statistics test are repectively equal to 80 and 2.5, whereas in the pre calculus test they are respectively equal to 70 and 2. if vince got a score of 85 in statistics and a score of 75 in pre-calculus, in which subject is her standing better assuming normality on both subject?

Answer 1

$\mu _1=80, \sigma _1=2.5\\P\left( X_1\leqslant 85 \right) =P\left( \frac{X_1-\mu _1}{\sigma _1}\leqslant \frac{85-\mu _1}{\sigma _1} \right) =P\left( Z\leqslant \frac{85-80}{2.5} \right) =\varPhi \left( 2 \right) \\\mu _2=70,\sigma _2=2\\P\left( X_2\leqslant 75 \right) =P\left( \frac{X_2-\mu _2}{\sigma _2}\leqslant \frac{75-\mu _2}{2} \right) =P\left( Z\leqslant \frac{75-70}{2} \right) =\varPhi \left( 2.5 \right) \\Since\,\,\varPhi \left( 2 \right) <\varPhi \left( 2.5 \right) , in\,\,pre-calculus\,\,Vince\,\,is\,\,in\,\,higher\,\,percentile\\$