In a recent year, the ACT scores for the reading portion of the test were normally distributed, with a mean of 21.3 and a standard deviation of 6.2. Find the probability that a randomly selected high school student who took the reading portion of the ACT has a score that is (a) less than 15, (b) between 18 and 25, and (c) more than 34, and (d) identify any unusual events. Explain your reasoning

a)

And we can find this probability using the normal standard table or excel and we got:

b)

And we can find this probability with this difference:

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.  

c)

And we can find this probability using the complement rule and the normal standard table or excel and we got:

d)

If a value is lower than 8.9 or higher than 33.7 would be considered as unusual.

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the scores of a population, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability using the normal standard table or excel and we got:

Part b

And we can find this probability with this difference:

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.  

Part c

And we can find this probability using the complement rule and the normal standard table or excel and we got:

Part d

For this case we can use the rule of thumb that we expect about 95% of the data values between two deviations and we can find the normal limits like this:

If a value is lower than 8.9 or higher than 33.7 would be considered as unusual.