Answer:
The probability that a random selected student score is greater than 76 is
.
Step-by-step explanation:
The Normally distributed data are described by the normal distribution. This distribution is determined by two <em>parameters</em>, the <em>population mean</em>
and the <em>population standard deviation</em>
.
To determine probabilities for the normal distribution, we can use <em>the standard normal distribution</em>, whose parameters' values are
and
. However, we need to "transform" the raw score, in this case <em>x</em> = 76, to a z-score. To achieve this we use the next formula:
[1]
And for the latter, we have all the required information to obtain <em>z</em>. With this, we obtain a value that represent the distance from the population mean in standard deviations units.
<h3>
The probability that a randomly selected student score is greater than 76</h3>
To obtain this probability, we can proceed as follows:
First: obtain the z-score for the raw score x = 76.
We know that:
![\\ \mu = 85](https://tex.z-dn.net/?f=%20%5C%5C%20%5Cmu%20%3D%2085)
![\\ \sigma = 3](https://tex.z-dn.net/?f=%20%5C%5C%20%5Csigma%20%3D%203)
![\\ x = 76](https://tex.z-dn.net/?f=%20%5C%5C%20x%20%3D%2076)
From equation [1], we have:
![\\ z = \frac{76 - 85}{3}](https://tex.z-dn.net/?f=%20%5C%5C%20z%20%3D%20%5Cfrac%7B76%20-%2085%7D%7B3%7D)
Then
![\\ z = \frac{-9}{3}](https://tex.z-dn.net/?f=%20%5C%5C%20z%20%3D%20%5Cfrac%7B-9%7D%7B3%7D)
![\\ z = -3](https://tex.z-dn.net/?f=%20%5C%5C%20z%20%3D%20-3)
Second: Interpretation of the previous result.
In this case, the value is <em>three</em> (3) <em>standard deviations</em> <em>below</em> the population mean. In other words, the standard value for x = 76 is z = -3. So, we need to find P(x>76) or P(x>-3).
With this value of
, we can obtain this probability consulting <em>the cumulative standard normal distribution, </em>available in any Statistics book or on the internet.
Third: Determination of the probability P(x>76) or P(x>-3).
Most of the time, the values for the <em>cumulative standard normal distribution</em> are for positive values of z. Fortunately, since the normal distributions are <em>symmetrical</em>, we can find the probability of a negative z having into account that (for this case):
Then
Consulting a <em>cumulative standard normal table</em>, we have that the cumulative probability for a value below than three (3) standard deviations is:
Thus, "the probability that a random selected student score is greater than 76" for this case (that is,
and
) is
.
As a conclusion, more than 99.865% of the values of this distribution are above (greater than) x = 76.
<em>We can see below a graph showing this probability.</em>
As a complement note, we can also say that:
![\\ P(z3)](https://tex.z-dn.net/?f=%20%5C%5C%20P%28z%3C-3%29%20%3D%201%20-%200.99865%20%3D%20P%28z%3E3%29)
![\\ P(z3)](https://tex.z-dn.net/?f=%20%5C%5C%20P%28z%3C-3%29%20%3D%200.00135%20%3D%20P%28z%3E3%29)
Which is the case for the probability below z = -3 [P(z<-3)], a very low probability (and a very small area at the left of the distribution).