Question

Among freshman at a certain university, scores on the Math SAT followed the normal curve, with an average of 550 and an SD of 100. Fill in the blanks; explain briefly. a) A student who scored 400 on the Math SAT was at the ______ th percentile of the score distribution. b) To be at the 75th percentile of the distribution, a student needed a score of about ______ points on the Math SAT.

Answer 1

Answer:

a) 6.68th percentile

b) 617.5 points

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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:

$\mu = 550, \sigma = 100$

a) A student who scored 400 on the Math SAT was at the ______ th percentile of the score distribution.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{400 - 550}{100}$

$Z = -1.5$

$Z = -1.5$ has a pvalue of 0.0668

So this student is in the 6.68th percentile.

b) To be at the 75th percentile of the distribution, a student needed a score of about ______ points on the Math SAT.

He needs a score of X when Z has a pvalue of 0.75. So X when Z = 0.675.

$Z = \frac{X - \mu}{\sigma}$

$0.675 = \frac{X - 550}{100}$

$X - 550 = 0.675*100$

$X = 617.5$