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ACT scores are distributed normally with mean 21 and standard deviation 5. What percent of scores fall between 28 and 19 on the ACT?\\  $z_{28} score=\frac{28-21}{5}=1.4$ with probability 0.9192. \\  $z_{19} score=\frac{19-21}{5}=-0.4$ with probability 0.3446\\  so the percent of scores between these valeus is values are  (0.9192-0.3446)100=57.46$\%$.\\ $\textbf{Example}$  you can also work back wards and find the observed value give nthe percentile.\\  If SAT scores are N(1500,300) and if sophire scored at 76th percentile, what was her actual score?\\  76th percentile has an associated z score to it so find that in the table. thats at z= 0.71. then using the z score equation you can solve for observed as, $oberserved=(300)(0.71)+1500=1713.$\\  $\textbf{Example}$\\  Let's assume SAT scores are ~ N(1500, 300). Between what two scores do the middle 50% of SAT test takers score?\\  In the probability curve we have the middle 50$\%$ that we want to cover. which means there is $\frac{.50}{2}$ left on each side of the curve. which means we need to no find the scores the way we did above but at .25 percentile and the .75 percentile.\\  \subsection{The Binomial Model}  The binomial probability distribution is a discrete probability distribution function\\  Useful in many situations where you have numerical variables that are counts or whole numbers\\  Classic application of the binomial model is counting heads when flipping a coin\\  The binomial model provides probabilities for random experiments in which you are counting the number of successes that occur. Four characteristics must be present:\\  \begin{itemize}  \item 1)Fixed number of trials: n  \item 2) The only two outcomes are success and failure  \item 3) The probability of success, p, is the same at each trial 4) The trials are independent  \end{itemize}