ACT mathematics score for a particular year are normally distributed with a mean of 28 and a standard deviation of 2.4 points

A. What is the probability a randomly selected score is greater than 30.4?

B. what is the probability a randomly selected score is less than 32.8?

C. What is the probability a randomly selected score is Between 25.6 and 32.8?

A) 0.1587

B) 0.9772

C) 0.8185

Step-by-step explanation:

A)

In this problem, the mathematics score of the year is distributed according to a normal distribution, with parameters:

is the mean of the distribution

is the standard deviation of the distribution

We want to find the probability that a randomly selected score is greater than 30.4. First of all, we calculated the z-score associated to this value, which is given by:

The z-score tables give the probability that the z-score is less than a certain value; since the distribution is symmetrical around 0,

Here we want to find , which is therefore equivalent to . Looking at the z-tables, we find that

B)

Here instead we want to find the probability that a randomly selected score is less than 32.8.

First of all, we calculate again the z-score associated to this value:

Now we notice that:

(1)

Since the overall probability under the curve must be 1. We also note that (from part A)

Which means that we can rewrite (1) as

Here, we have

Z = 2

This means that

Looking at the z-tables, we find that

Therefore, we get

C)

Here we want to find the probablity that the score is between 25.6 and 32.8.

First of all, we calculate the z-scores associated to these two values:

So here we basically want to find the probability that

Which can be rewritten as:

So in this case,

From part A and B we found that:

Therefore,