Download MATH-1410 5.1 Notes (6th Ed)

5.1 Introduction to Normal Distributions and
the Standard Normal Distribution
• Important Concepts:
– Normal Distribution
– Standard Normal Distribution
– Finding Areas Below the Standard Normal
Curve
5.1 Introduction to Normal Distributions and
the Standard Normal Distribution
• What is a normal distribution?
– A continuous probability distribution for a
random variable X.
– The graph of a normal distribution is called
the normal curve.
• The shape of a normal curve is determined by two
parameters – the mean and the standard deviation
of the random variable (see p. 235).
5.1 Introduction to Normal Distributions and
the Standard Normal Distribution
• If a variable is normally distributed, or follows a
normal distribution, what does that tell us?
– The mean, median, and mode of the variable are all
equal.
– The normal curve is bell-shaped and is symmetric
about the mean.
– The total area below the normal curve is 1.
– The tails of the normal curve are asymptotic to the xaxis.
– Inflection points are located one standard deviation
away from the mean.
5.1 Introduction to Normal Distributions and
the Standard Normal Distribution
• Examples of continuous random variables that
are normally distributed, i.e., follow a normal
distribution:
–
–
–
–
–
–
The heights and weights of adult males and females
The body temperatures of rats
The cholesterol levels of adults
Intelligence Quotients (IQ Scores)
Life spans of light bulbs
Weights of newborn infants
5.1 Introduction to Normal Distributions and
the Standard Normal Distribution
• The standard normal distribution is a normal
distribution with a mean of 0 and a standard
deviation of 1.
• Very important relationship we will need later:
– If X is a normally distributed variable with mean µ and
standard deviation σ, then the variable
z
x

follows the standard normal distribution.
5.1 Introduction to Normal Distributions and
the Standard Normal Distribution
• Finding areas below the standard normal curve.
– We use Table 4 to find cumulative areas
#18 p. 243 (area to the left)
#26* (area to the left)
#20 (area to the right)
#32 (area between two values)
– Standard normal curves and probabilities
#48 p. 245
#54
#56