Question

each year to many thousands of high school students, ore scored so as to yield mean of 500 and ฮ standard deviation of 100·These scores are close to eing normally distributed. An exclusive dub wishes to invite those scoring in the top 10% on the College Boards to join. (a) What score is required to be invited to join the club? or greater (b) what iccre separates the top 70% of the population from the bottom 30%? What do we cal this value? O This is the 701h percentle O This is the 30th percentile O This is the median.

Answer 1

Solution:

We are given:

$\small \mu=500, \sigma=100$

We are given that an exclusive club wishes to invite those scoring in the top 10% on the college board to join.

We have to find the score which is required to be invited to join the club. In other words, we have to find the score which has 10% of the scores above it and 90% of the scores below it, that is, we need to find the z value corresponding to area 0.90. Using the standard normal table, we have:

$z(0.90)=1.28$

Now using the z-score formula, we have:

$z=\frac{x-\mu}{\sigma}$

$1.28=\frac{x-500}{100}$

$128=x-500$

$x=500+128$

$x=628$

Therefore, the score that is required to be invited to join the club is $\boldsymbol{628}$ or greater

(b) What score separates the top 70% of the population from the bottom 30%?

Answer:

We first need to find the z value corresponding to the area 0.3. Using the standard normal table, we have:

$z(0.3)=-0.52$

Now using the standard normal table, we have:

$z=\frac{x-\mu}{\sigma}$

$-0.52=\frac{x-500}{100}$

$-0.52=x-500$

$x=500-52$

$x=448$

Therefore, the score which separates the top 70% of the population from the bottom 30% is $\boldsymbol{448}$.

What do we call this value?

Answer: This is the 30^th percentile.