The marks on a statistics exam are normally distributed with mean 70 and standard deviation 10
- Looking at this problem first you should notice that it asks you for statistics exam grade being
“Normally distributed” That means that the graph of a normal distribution will be normal Bell Curve.
- This problem gives a Mean and Standard Deviation
Mean gives the location of the line of symmetry (in the middle) and standard deviation describes how much the data is spread out
- In this problem we know that:
Mean (µ) =70 SD=10
The Formula we need to use in this problem is Normal Distribution formula Z=X-M/Q
A) What proportion of students will receive more than 80?
X=80, M=70, Q=10 80 -70/10= -1 Z= 1 After we find Z “1.000” we need to find its Area in Standard normal Distribution Table; you will find that 1=.8413 You can see that X=80 is > than our mean 70 and is located on the right side of our Bell Curve – which means we have to subtract 1 from our Z score area
1 – .8413=.1587 Our Final Answer
B) Find the probability that a mark will be between 60 and 90
We can see that our two X’s are 60 and 90 therefore we have to apply both numbers separately to our Normal Distribution Formula.
60–70/10= -1
90-70/10=2
Now that we found Z scores for both our X’s we have to look them up in Standard normal Distribution table for its Area’s
-1=.1587
2=.9772 Because 90 is located on the right of our Bell Curve we have to subtract 1 from our Area 1-.9772=.0228
To find the probability in between of 60 and 90 we have to subtract bigger Z score from Smaller
.1587-.0228=.1359 Our Final Answer
C) If less than 60 is a failing grade, what is probability that a student fails the class
60–70/10= -1
Now that we found the Z score for both our X’s we have to look them up in Standard normal Distribution table for its Area’s
-1=.1587 Our Final Answer
D) If only the best 10% of the grade in the class will receive A, what grade must a student obtain in order to get an A?
10%=.10 From the problem they ask if only the “BEST” from the class will receive A, we can assume that its more than our M=70 which will be located to the right of our Bell Curve
Because it’s on the right we automatically subtract 1 from .10
1-.10=.9 Find the Z value using the in Standard normal Distribution table
The closest value to .9 is .8997 Therefore Z=1.28
Now we have to convert z to x using Normal Distribution formula”
X=M+ZQ
1.28*10=12.8+70=82.8 Our Final Answer
