Average Math SAT score and an overachieving high school- Statistics, std, Dev help

Question

the average math SAT score is 517 with a standard deviation of 118. A particular high school claims that its students have unusually high math SAT scores. A random sample of 55 students from this school was selected, and the mean math SAT score was 561. Is the high school justified in its claim?

Answer 1

We need to find out whether 561 deviates significantly from the mean 517 to be considered exceptionally high. Work out the score (561-517)/118=0.3729 approximately. As a Z score this corresponds to 0.645, which means that 64.5% of SAT results would be below 561 and 35.5% would therefore be above 561. To be significant, we need only a small number of results to be higher than 561 (say, 5%) if we were to consider 561 to be exceptionally high. Even using the t-test with 55-1=54 degrees of freedom, we are nowhere near the critical value, so we can say with a reasonable degree of confidence that 561 is not significantly high enough to be considered exceptional, so the high school is not justified in its claim.