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Math 166, Fall 2015, Robert
Williams
3.4- The Normal Distribution
Continuous random variables cannot be worked with in the same manner as
discrete variables. It is impossible to draw histograms for continuous variables.
However, the way to calculate probability remains the same: we find the area
under the curve of the probability distribution function.
We will focus on the normal distribution:
Characteristics of the Normal Curve:
• It is bell shaped
• It is symmetric about x = µ
• It lies entirely above the x-axis (i.e. it never touches or crosses the x-axis)
• It approaches, but never equals, 0 as x goes to ∞ and −∞
• The curve bends downward on the interval (µ−σ, µ+σ) and bends upward
outside of this interval
• The area under the entire curve is 1
The standard normal curve or standard normal distribution is a normal curve
with µ = 0 and σ = 1
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Math 166, Fall 2015, Robert
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Calculating probabilities that have a normal distribution requires knowledge of
calculus, so we will rely on built-in calculator functions. For a normal distribution with mean µ and standard deviation σ:
• normalcdf(a, b, µ, σ) calculates P (a < X < b)
– When we want P (X < b), we will use a = −1 × 1099 . When we want
P (a < X), we will use b = 1 × 1099
• invNorm(p, µ, σ) calculates the value of A where P (X < A) = p
Example 1 At the end of a course, a teacher notices that the students’ grades
are normally distributed with a mean of 64 and a standard deviation of 13. The
teacher decides to rescale the grade scale to reflect the difficulty of the course.
1. If the teacher wants 5% of the students to make an F, what should be the
highest average that gets an F?
2. If the teacher wants 10% of the students to make an A, what should be the
cut off grade for it?
3. If the teacher wants 20% of the students to make a B, what should be the
cut off grade for it?
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Math 166, Fall 2015, Robert
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Example 2 A manufacturer produces bolts to be used in the construction of
other machines. The diameter of one particular bolt that is produced is normally
distributed with a mean of 10 mm and a standard deviation of 0.003 mm.
• What is the probability that a bolt produced will have a diameter smaller
than 9.995mm?
• Bolts are considered defective if their diameter is not in the interval [9.99, 10.01].
What is the probability of producing a defective bolt?
Example 3 A cell phone manufacturer records data on how long it takes their
products to break once purchased. They find that the amount of time until the
device breaks is distributed normally with a mean of 4 years and a standard
deviation of 0.735 years.
1. What is the probability that a phone produced by this company will last for
at least 3 years, but no more than 5?
2. If the manufacturer wants to ensure that 98% of their phones will not break
during the warranty period, how long of a warranty should they offer?
Example 4 Let Z be the standard normal variable. Find the value of a that
satisfies P (−a < Z < a) = .4
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