Download Section 7.1 – Properties of the Normal Distribution

Section 7.1 – Properties of the Normal Distribution
Objectives
1. Use the uniform probability distribution
2. Graph a normal curve
3. State the properties of the normal curve
4. Explain the role of area in the normal density function
Objective 2 – Graph a Normal Curve
Probability Density Function – An equation used to compute probabilities of continuous random
variables. It must satisfy the following 2 properties:
1. Total area under the curve must equal 1.
2. The height of the graph must be greater than or equal to 0 for all possible values of the
random variable.
Note the name change. For discrete random variables we called it a probability distribution function.
For continuous random variables we call it a probability density function.
Relative frequency histograms that are symmetric and bell-shaped are said to have the shape of a
normal curve.
Normal Distribution – A continuous random variable is normally distributed if the relative frequency
histogram is symmetric and bell-shaped.
Objective 3 – State the Properties of the Normal Curve
Properties of the Normal Density Curve
1. Symmetric about the mean, μ.
2. Mean = Median = Mode, so there is a single peak and the highest point occurs at x = μ.
3. Inflection points occur at μ – σ and μ + σ.
4. The area under the curve = 1.
5. The areas to the right and left of μ are both equal to ½.
6. As x increases and decreases without bound, the graph approaches (but never touches) the
horizontal axis.
7. The Empirical Rule applies:
a. Approximately 68% of the area under the curve is between μ – σ and μ + σ
b. Approximately 95% of the area under the curve is between μ – 2σ and μ + 2σ
c. Approximately 99.7% of the area under the curve is between μ – 3σ and μ + 3σ
(see pg. 364)
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The normal curve is a mathematical model (an equation, table, or graph that is used to describe
reality). The normal curve does a good job of describing the distribution of things like height, IQ
scores, cholesterol, birth weights, etc.
An inflection point is where the graph changes concavity.
Objective 4 – Explain the Role of Area in the Normal Density Function
Finding the area under the normal curve is equivalent to finding the probability corresponding to that
random variable. Or put another way, the area under the graph of the density function over an
interval represents the probability of observing a value of the random variable in that interval.
The area under the normal curve = Probability
Area under a Normal Curve
Suppose that a random variable X is normally distributed with mean μ and standard deviation σ. The
area under the normal curve for any interval of values of the random variable X represents either the
 proportion of the population with the characteristic described by the interval of values or
 probability that a randomly selected individual from the population will have the
characteristic described by the interval of values
Example
The lives of refrigerators are normally distributed with mean  = 14 and standard deviation  == 2.5
years.
a) Draw a normal curve with the parameters labeled.
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b) Shade the region that represents the proportion of refrigerators that last for more than 17
years.
c) Suppose the area under the normal curve to the right of x = 17 is 0.1151. Provide two
interpretations of this result.
Example: Review problem 38, page 369, in StatCrunch.
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Section 7.2 – Applications of the Normal Distribution
Objectives
1. Find and interpret the area under a normal curve
2. Find the value of a normal random variable
The two calculator functions used in this section are
2nd DISTR
normalcdf(lower bound, upper bound, µ, σ)
and
2nd DISTR
invNorm(area to left, µ, σ)
Objective 1 - Find and interpret the area under a Normal Curve:
There are two cases to consider:
1. A standard normal distribution
2. A normal distribution that is not standard
A standard normal distribution is a normal distribution that has mean 0 and standard deviation 1.
Any normal random variable can be converted to a standard normal random variable by using Zscores.
By converting to a standard normal distribution we can use Table V instead of a calculator to find the
area under the curve, which of course is the probability. The TI 83/84 calculator handles any normal
distribution, standard or not.
Also, the area under the curve to the right of z = 1 – the area under the curve to the left of z.
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Figure 1 Standard Normal Curve
Figure 2 Areas Under the Standard Normal Curve
Z-score:
The distance along the horizontal scale of the normal distribution.
It counts how many standard deviations above or below the mean a data point is.
Z-scores are rounded to two decimals
Area:
The region underneath the curve. It is probability!
Area/Probability is rounded to four decimals
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Finding the Area under a Normal Curve using Table V:
1. Write the inequality statement for X
2. Convert the values of X into Z-scores
3. Draw a standard normal curve and shade the desired area
4. Find the area under the curve using Table V
Finding the Area under a Normal Curve using TI 83/84:
2nd DISTR
normalcdf (lower bound, upper bound, µ, σ)
The output on your calculator will be an AREA, so round it to 4 decimals
For the standard normal distribution: µ = 0 and σ = 1
For -∞ (negative infinity) enter
–1E99
For ∞ (positive infinity) enter
1E99
Note: The lower bound must be listed first before the upper bound.
Example:
Use Table V at the back of the book (pg A-11 and A-12) to find the area under the standard normal
curve between z = –1.02 and z = 2.94.
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Example: Finding area under the standard normal curve that lies to the LEFT of a z-score
P (Z < 2.29) = normalcdf (–1E99, 2.29)
Example: Finding area under the standard normal curve that lies to the RIGHT of a z-score
P (Z > –1.02) = normalcdf (–1.02, 1E99)
Example: Finding area under the standard normal curve that lies BETWEEN two z-scores
P (–0.12 < Z < 1.02) = normalcdf (–0.12, 1.02)
Example:
It is known that the length of a certain steel rod is randomly distributed with a mean of 100cm and a
standard deviation of 0.45cm. What is the probability that a randomly selected steel rod has a length
less than 99.2cm?
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It is known that the length of a certain steel rod is randomly distributed with a mean of 100cm and a
standard deviation of 0.45cm. What is the probability that a randomly selected steel rod has a length
between 99.8 and 100.3cm?
It is known that the length of a certain steel rod is randomly distributed with a mean of 100cm and a
standard deviation of 0.45cm. Suppose it is known that the manufacture must discard all rods less
than 99.1cm and longer than 100.9cm. What proportion of rods must be discarded?
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Objective 2 - Find the value of a Normal Random Variable
In the section we are given an area under the normal curve and must find the corresponding
normal value.
Calculator Instructions
For the standard normal distribution: µ = 0 and σ = 1
The output on your calculator will be a Z-score so round it to 2 decimals
Examples:
Find the z-score so that the area under the standard normal curve to its left is 0.7157.
For the standard normal curve, find the z-score that is the cutoff for P70.
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Find the z-score so that the area under the standard normal curve to its right is 0.9656.
Find the z-score that separates the middle 92% of the area under the standard normal curve from the
area in the tails.
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Finding the Value of a Normal Random Variable that Corresponds to a Given Proportion,
Probability, or Percentile
1. Draw a standard normal curve and shade the area corresponding to the given
probability or percentile.
2. Find the Z-score that corresponds to the shaded area using “invNorm” or Table V.
3. Convert the Z-score back to a data point using the formula x =   z .
-or2nd DISTR
invNorm (area to the LEFT of the z-score, µ, σ)
Example: The combined (verbal + quantitative reasoning) score on the GRE is normally distributed
with a mean of 1049 and standard deviation of 189. What is the score of a student whose percentile
rank is at the 85th percentile, rounded to the nearest whole number?
Example: It is known that the length of a certain steel rod is normally distributed with a mean of
100cm and a standard deviation of 0.45cm. Suppose the manufacturer wants to accept 90% of all
rods manufactured. Determine the length of the rods that make up the middle 90% of all steel rods
manufactured.
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Notation for later in the course
z (pronounced z sub-alpha) is the z-score such that the area under the standard normal curve to the
right of z is  .
Example: Find the value of z0.25
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