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Chapter 5

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Statistical Sciences

Statistical Sciences 2244A/B

Jennifer Waugh

Spring

Description

5.1 Overview
● Arandom variable is a variable having a single numerical value, determined by chance,
for each outcome of some procedure
● Aprobability distribution for a discrete random variable describes the probability for each
value of the random variable
● Adiscrete random variable has either a finite number of values or a countable number of
values. That is, the number of possible values that x can assume is 0, or 1, or 2, etc.
● Acontinuous random variable has infinitely many values and those values are often
associated with measurements on a continuous scale with no gaps or interruptions
● In this chapter, continuous probability distributions are presented
● If a continuous random variable has a distribution with a graph that is symmetric and
bell-shaped, and can be described by the equation given, we say it has a normal
distribution
● The normal distribution is determined by two parameters: the mean, μ, and standard
deviation, σ
5.2 The Standard Normal Distribution
Uniform Distributions
● Acontinuous variable has a uniform distribution if it’s values spread evenly over the
range of possibilities; the graph of a uniform distribution results in a rectangular shape
● The graph of a discrete probability distribution is called a probability histogram; the
graph of a continuous probability distribution is called a density curve
○ density curve must satisfy the following properties:
■ total area under the curve must equal 1
■ every point on the curve must have a vertical height that is 0 or greater
● Due to the total area under the density curve equalling 1, there is a correspondence
between the area and probability
Standard Normal Distributions
● The density curve of uniform distributions is rectangular, thus easy to find
areas/probabilities but for a density curve of normal distribution, it is more complicated
due to the bell shape
○ despite the shape, correspondence between area and probability remains
● The standard normal distribution is a normal probability distribution that has a mean of 0
and a standard deviation of 1, and the total area under its density curve is equal to 1
○ Using the standard normal distribution is easiest for calculations
Finding Probabilities When Given z Scores
● z score: the distance along the horizontal scale of the standard normal distribution
● Area: region under the curve
● Notation:
○ P(a < z < b) - denotes probability that z score is between a and b
○ P (z > a) denotes probability that z score is greater than a
○ P (z < a) denotes probability that z score is less than a
● With a continuous probability distribution such as a normal distribution, the probability
of getting any single exact value is 0; P(z=a) = 0 ○ e.g. there is a 0 probability of randomly selecting someone and getting a height of
68.12345667 in.
○ In a normal distribution, any single point on a horizontal scale is represented not
by a region under the curve, but by a vertical line above the point; a line with no
area
● P(a <= z <= b) = P(a < z < b)
Finding z Scores from KnownAreas
● Note, z scores positioned in the left half of the curve are always negative
● In cases where we know the area or probability and want to find the corresponding z
score, remember to avoid confusion between z scores and areas and following the steps:
○ Draw a bell shaped curve and identify region under the curve that corresponds to
given probability
○ Using cumulative area from left, locate the closest probability in the body of
TableA-2 and identify the corresponding z score
● Note, when finding the z score and a desired value is midway between two table values,
select the larger value to be safe
5.3 Applications of Normal Distributions
● Standard normal distributions with means or 0 and standard deviations of 1 are unrealistic
● However, when working with nonstandard normal distributions, fortunately we can
transform its values to a standard normal distribution
● If we convert values to standard scores, then procedures for working with all normal
distributions are same as those for the standard normal distribution
○ z= (x - μ) / σ
● The area in any normal distribution bounded by some score x is the same as the area
bounded by the equivalent z score in the standard normal distribution
Finding Values from KnownAreas
● In many practical and real cases, the area (probability) is known and we must find the
relevant value(s); when finding them, be cautious of the following:
○ Don’t confuse z scores and areas
○ Choose the correct side of the graph (right/left)
○ Az score must be negative whenever it is located in the left half of the normal
distribution
○ Areas (probabilities) are positive or zero values but they are never negative
● Remember, when rearranging the equation z= (x - μ) / σ into x = μ + (z * σ) to solve for
the value x from a known area (and thus known z score), if the z is located to the left of
the mean, be sure it is a negative number
5.4 Sampling Distributions and Estimators
● The main objective of this section is to learn what we mean by a sampling distribution of
a statistic, and another is to learn a basic principle about sampling distribution of sample
means and the samplingt distribution of sample proportions
● It is rare that we know

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