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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:
From equation [1], we have:
Then
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:
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).
