Question

The scores of high school seniors taking the ACT college entrance examination in 2003 was, mean= 19.6 with standard deviation of 3.8. assume the distributing of the scores to be roughly
What is the probability that a single student chosen a random from all those taking the test scored 22 or higher?
Assume that the problem was to calculate the probability of randomly selected student scoring 23 or higher. would you expect the probability to increase, decrease, or remain the same?

Answer 1

Denote by $\xi$ random variable that corresponds to the scores of high school seniors taking the ACT college entrance. We know that $\xi$ has roughly normal distribution with mean equal to 19.6 and standard deviation of 3.8. Thus,

$$\xi \sim N(19.6, 3.8)$$

We need to find probability that a single student chosen at random is taking the test with score 22 or higher.

$$P(\xi \geq 22)$$

Using properties of normal distribution we have:

$$P(\xi \geq 22) = 1 - P(\xi < 22) = 1 - \Phi\left(\frac{22 - 19.6}{3.8}\right) = 0.2638$$

Thus this probability equals 0.2638.

If we look for probability of scoring 23 or higher then

$$P(\xi \geq 23) < \{\text{using monotonicity of probability}\} < P(\xi \geq 22)$$

Thus, in this case probability decreases.