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:
[tex]\mu=28[/tex] is the mean of the distribution
[tex]\sigma = 2.4[/tex] 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:
[tex]z=\frac{30.4-\mu}{\sigma}=\frac{30.4-28}{2.4}=1[/tex]
The z-score tables give the probability that the z-score is less than a certain value; since the distribution is symmetrical around 0,
[tex]p(z>Z) = p(z<-Z)[/tex]
Here we want to find [tex]p(z>1)[/tex], which is therefore equivalent to [tex]p(z<-1)[/tex]. Looking at the z-tables, we find that
[tex]p(z<-1)=0.1587[/tex]
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:
[tex]z=\frac{32.8-\mu}{\sigma}=\frac{32.8-28}{2.4}=2[/tex]
Now we notice that:
[tex]p(z<Z) = 1-p(z>Z)[/tex] (1)
Since the overall probability under the curve must be 1. We also note that (from part A)
[tex]p(z>Z) = p(z<-Z)[/tex]
Which means that we can rewrite (1) as
[tex]p(z<Z) = 1-p(z<-Z)[/tex]
Here, we have
Z = 2
This means that
[tex]p(z<2)=1-p(z<-2)[/tex]
Looking at the z-tables, we find that
[tex]p(z<-2)=0.0228[/tex]
Therefore, we get
[tex]p(z<2)=1-0.0228=0.9772[/tex]
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:
[tex]z_1=\frac{25.6-\mu}{\sigma}=\frac{25.6-28}{2.4}=-1[/tex]
[tex]z_2=\frac{32.8-\mu}{\sigma}=\frac{32.8-28}{2.4}=2[/tex]
So here we basically want to find the probability that
[tex]p(z_1 <z<z_2)[/tex]
Which can be rewritten as:
[tex]p(z_1<z<z_2)=1-p(z<z_1)-p(z>z_2)[/tex]
So in this case,
[tex]p(-1<z<2)=1-p(z<-1) -p(z>2)[/tex]
From part A and B we found that:
[tex]p(z<-1)=0.1587[/tex]
[tex]p(z>2)=1-p(z<2)=1-0.9772=0.0228[/tex]
Therefore,
[tex]p(-1<z<2)=1-0.1587-0.0228=0.8185[/tex]
