Scores on the SAT form a normal distribution with a mean of µ = 500 with σ = 100. If the state college only accepts students who score in the top 60% on the SAT, what is the minimum score needed to be accepted?

Answer:The minimum score needed to be accepted is 525.3.Step-by-step explanation:Problems of normally distributed samples are solved using the z-score formula.In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:[tex]Z = \frac{X - \mu}{\sigma}[/tex]The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.In this problem, we have that:[tex]\mu = 500, \sigma = 100[/tex]If the state college only accepts students who score in the top 60% on the SAT, what is the minimum score needed to be accepted?The 60th percentile, which is the value of X when Z has a pvalue of 0.6. So it is X when Z = 0.253. So[tex]Z = \frac{X - \mu}{\sigma}[/tex][tex]0.253 = \frac{X - 500}{100}[/tex][tex]X - 500 = 0.253*100[/tex][tex]X = 525.3[/tex]The minimum score needed to be accepted is 525.3.