In some state the mean ACT score in Mathematics is mu_ACT = 19 and the standard

Question

In some state the mean ACT score in Mathematics is mu_ACT = 19 and the standard deviation is sigma_ACT = 5, while the mean SAT score in Mathematics is mu_ACT = 480 and the standard deviation is sigma_SAT = 110. Suppose that the ACT scores and the SAT scores have a normal distribution. Terence gets a score of 13 on the Mathematics of the ACT and Philip gets a score of 343 on the Mathematics portion of the SAT. Answer questions (a)-(b) using the Empirical Rule. You should not use R here since the point of this exercise is to practice the Empirical Rule. What proportion of ACT scores are less than 9? What proportion of SAT scores are greater than 810? What proportion of ACT scores are between 14 and 29? What proportion of SAT scores are between 370 and 590? What is the z-score of Terence? What is the z-score of Philip? Who did better in Mathematics, Terence or Philip?

Explanation / Answer

Part-c

According to empirical rule for 1 SD, we have

P(ACT<14)=(1-0.68)/2

=0.16

Also empirical rule for 2 SD states that we have

P(ACT>29)=(1-0.95)/2

=0.025

So, P(14<ACT<29)=1-0.16-0.025=0.815

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