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IB Math SL Year 2
Name__________________________
Date___________________________
3-5: Normal Distribution
Learning Goal: What is normal distribution and how do we answer questions that use the normal curve?
The Normal Model:


The Normal Model is “bell-shaped”, symmetric, and
unimodal
The normal model summarizes the shape of a continuous,
quantitative distribution
 Often, you will see the notation: 𝑋~𝑁(𝜇, 𝜎)
N - Normal model
μ - population mean
σ2 - variance


Centered around the mean
Standard deviation measures the spread of the data
Consider the model to the right. Which variable 𝑋1 𝑋2 , 𝑜𝑟 𝑋3 ,has the largest
standard deviation?
A percentile is
_____________________________________________________________
a. For each of the following identify the percentile being discussed, sketch a normal curve to show the given
percentage:
a) Someone being in the top 10% of your class
b) Someone who is in the lowest 5% for their height
c) Someone who is in the top 60% for their weight
IB Math SL Year 2
Normal Distribution
Normal
Just READ! The lengths of 100 pipes have a normal
distribution with a mean of 102.4 inches and a standard
deviation of 0.2 inch. Find the probability that the pipes
lie between 98 and 100 inches.
Inverse Normal
Just READ! The marks of 500 candidates in an exam are
normally distributed with a mean of 55 marks and a
standard deviation of 15 marks.
If 5% of the candidates obtained distinction by scoring d
marks or more, find the value of d.
How are these two examples different from each other?
Q: How can we use our calculator to determine a
probability about data that is approximately normal?
Q: How can we use our calculator to determine a
value in approximately Normal data?
A: When asked for a percentage/probability of being
above/below/between a value(s) in a Normal Model
follow these steps!
A: When asked for a value and you are given a
percentage along with a Normal Model follow these
steps!
1. Draw and label the normal curve
1. Draw and label the normal curve
a. Shade the curve in the proper direction based off what
probability you want to find
b. More than means shade above or to the right
c. Less than means shade below or to the left
d. Between means shade between the two given values.
2. Hit 2nd then VARS to access the DISTR menu
Use normalcdf(low, high, mean, st.dev)
If you only have 1 bound use
_________________for the LOW
value OR ________________ for the
HIGH value
Now solve: the lengths of 100 pipes have a normal
distribution with a mean of 102.4 inches and a
standard deviation of 2 inches. Find the probability
that the pipes lie between 98 and 100 inches.
2. Use invnorm(percentile of the missing value,
mean, st.dev)
Percentiles measure the
percent BELOW.
Ex: If you are in the 90th
percentile for your height that
means 90% of people are
shorter than you!
Now solve: The marks of 500 candidates in an exam are
normally distributed with a mean of 55 marks and a
standard deviation of 15 marks. If 5% of the candidates
obtained distinction by scoring d marks or more, find the
value of d.
IB Math SL Year 2
Another way to look at it…
Let X be the random variable such that, 𝑋~𝑁(10, 22 ). Determine 𝑃(9.1 < 𝑋 < 10.3)
Will we use normalcdf or invnorm? How do you know? ____________________________________________________________
𝜇= ______
𝜎 = ______
You try…
𝜇= ______
𝜎 = ______
Another type…
Determine the value for a with 𝜇=2.6 and 𝜎 = 0.05.
Will we use normalcdf or invnorm? How do you know? ____________________________________________________________
Now, solve:
Error Analysis! What is wrong with Joe’s work below? Identify the error and correct it!
Mensa is an organization whose members possess IQs in the top 2% of the population. If the IQs are normally
distributed with a mean of 100 and a standard deviation of 15, what is the minimum IQ necessary for admission?
IB Math SL Year 2
One more together!
a) F has a normal distribution with mean 100 and standard deviation of 25.
Find P(|F-100|<15)
b) F has a normal distribution with mean 100 and standard deviation of 25.
Find P(|F-100|>10)
5-5 Practice
1. Find P(Y>5.1, if Y~M(4.8, 1.44).
IB Math SL Year 2
2. On a Physics test, the distribution of scores is normal, the mean of the scores
is 75, and the standard deviation is 5.8.
a) Find the probability the scores would fall below 65
b) The top 15% of kids scored what on this test?
c) The lowest 5% earned what score or below?
d) What’s the probability that scores fall between a 70 and 83?
e) What’s the probability someone scores above a 90?
6. If X~N(14, 49), find the value of x for which P(X<x) =0.08.
IB Math SL Year 2
7. Given that X~N(16,2.52 ), find, P(12.5<x<16.5)
8. A set of normally distributed student test scores has a mean of 80 and a standard deviation of 4.
a. Determine the probability that a randomly selected score will be between 74 and 82.
b. In this group of students, 8% have a score greater than s. Find the value of s.
9. Given that Z~N(0,1), findP(-2 < Z < 2).
Mean: _______
sd dev: ________
10. Suppose scores on an IQ test are normally distributed. If the test has a mean of 100 and a standard deviation of 10.
a) What IQ must I have to be in the 50th percentile? Explain.
b) What IQ must I have to qualify as the smartest 10% of people? Use a picture as
part of your explanation.
IB Math SL Year 2
11. Regulations in a country insist that all mineral bottles that claim to contain 500ml must have at least that amount.
“Yummy Cola” has a machine for filling bottles, which puts an average of 502ml into each bottle with a standard
deviation 1.6ml and follows a normal distribution.
a. An inspector randomly selects a bottle of “Yummy Cola.” What is the probability that it will break the
regulations?
b. What proportion of bottles will contain between 500ml and 505 ml?
c. 95% of bottles contain between a ml and b ml of liquid where a and b are symmetrical about the mean.
What are a and b?
12. A machine produces bolts with diameters normally distributed with a mean of 4mm and a standard deviation of
0.25mm. Bolts are measured accurately and any which are smaller than 3.5mm or bigger than 4.5 mm are rejected.
Out of a batch of 500 bolts, how many would be acceptable?