Question

The results of a state-wide examination of the reading skills of sixth-grade students were

normally distributed, with a mean score of 104 and a standard deviation of 16. The

students in the top 10% are to receive an award, and those in the bottom 14% will be

required to take a special reading class.

a. What score does a student need in order to receive an award?

b. What is the cut-off score that will be used to determine whether a student will be

required to take the special reading class?

Answer 1

Consider that the mean of the grades is 104 and the standard deviation is 16. The students in the top 10% are to receive an award, and those in the bottom 14% will be required to take a special reading class.

a. The following figure shows a standard normal distribution with 10% of the data to the right of some unknown z-score and 40% of the data to the left of the z-score but to the right of the mean of 0.

Hence, the unknown z-score is right of 0, thus

$z_x = 1.28$

Let x represents the score for which a person receive an award, then x is related to the z-score by the formula

$z_x = \frac{x -\bar{x}}{s} \\ 1.28 = \frac{x-104}{16} \\ 20.48 = x-104 \\ x = 124.48$

Hence, a student needs a score of 124.48 in order to receive an award.

b. The following figure shows a standard normal distribution with 14% of the data to the left of some unknown z-score and 36% of the data to the right of the z-score but to the left of the mean of 0.

Hence, the unknown z-score is to the left of 0, thus

$z_x = -1.08$

Let x represents the score for which a person receive an award, then x is related to the z-score by the formula

$z_x = \frac{x -\bar{x}}{s} \\ -1.08 = \frac{x-104}{16} \\ -17.28 = x -104 \\ x = 86.72$

Hence, the cut-off score to determine the requirement of special reading class is 86.72