Question

The normal distribution has a mean of 140 and a standard deviation of 40. How do you calculate the percentile rank of a score of 172?
 (a) 70%
(b) 65%
(c) 80%
(d) None of these

Answer 1

Hint: From the problem, we can see that the normal distribution has a mean of 140 and a standard deviation of 40. We are trying to find the percentile score of 172. So, to find the value, we will start with the given formula of $z=\dfrac{x-\mu }{\sigma }$. Then by putting the values and calculating in a simplified form will give us the result.

Complete step by step solution:
According to the question, the normal distribution has a mean of 140 and a standard deviation of 40. We need to know how we calculate the percentile rank of a score of 172.
The question requires the need of z-scores.
The formula of the needed one is, $z=\dfrac{x-\mu }{\sigma }$ .
Where we have, x = the given value, $\mu $ = the mean, $\sigma =$ the standard deviation.
So, it can be written as, $z=\dfrac{the\,given\,value-the\,actual\,mean}{SD}$
This is a bit easier to remember, where SD denotes the standard deviation.
Now, we will put the values from the given question.
Putting the values, $z=\dfrac{172-140}{40}$
Simplifying, $z=\dfrac{32}{40}=0.8$
Now, we are to use the percentage value of the given result. So, we will multiply the value we got with 100 and thus we will get our final result. The probability that corresponds to the z score is, $0.8\times 100=80%$ .

So, the correct answer is “Option C”.

Note: Here we have our given answer as 80% as per to the given values to us. Now, it can also be the case that we might have to check the probability that corresponds to the z score in the table. (Which we should be provided with.) And that will definitely give us a different result.