Answer to Question #315149 in Statistics and Probability for rainyyyyy

Question

The graduate record exam is standardized test recquired to be admitted to many graduate schools in the United States. A high score in GRE makes admissions more likely. According th the educational testing service, the mean score for takers of GRE who do not have training courses is 555 with standard deviation of 139. Brain Philippine offers expensive GRE training courses, claiming their graduates score better than those who have not talen any training courses. To test the company's claim, a statisticiam randomly selected 30 graduates of BP and asked their GRE scores. 

Suppose the desicion rule is “Reject Ho if the mean score of the sample d BP graduates is greater than 590 otherwise, accept Ho". Compute for the level of significance for this test. Also, find the risk of concluding that the BP graduates did not score better than 555 when in fact the mean score is 600. Use 95% confidence level.

Answer 1

$\alpha =P_{H_0}\left( \bar{x}>590 \right) =P_{H_0}\left( \sqrt{n}\frac{\bar{x}-\mu}{\sigma}>\sqrt{n}\frac{590-\mu}{\sigma} \right) =P\left( Z>\sqrt{30}\frac{590-555}{139} \right) =\\=P\left( Z>1.37916 \right) =\varPhi \left( -1.37916 \right) =0.0839\\\beta =P_{H_1}\left( \bar{x}\leqslant 590 \right) =P_{H_1}\left( \sqrt{n}\frac{\bar{x}-\mu _1}{\sigma}\leqslant \sqrt{n}\frac{590-\mu _1}{\sigma} \right) =P\left( Z\leqslant \sqrt{30}\frac{590-600}{139} \right) =\\=P\left( Z\leqslant -0.394045 \right) =\varPhi \left( -0.394045 \right) =0.3468$