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## Normal Probability Distributions

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Title: Normal Probability Distributions

1
Normal Probability Distributions
2
Overview
3
Normal Distribution
• If a continuous random variable has a
distribution with a graph that is symmetric and
bell-shaped, and it can be described by the
equation below, we say that it has a normal
distribution.

4
The Normal Distribution
• The curve is bell-shaped and symmetric.

5
The Standard Normal Distribution
6
Uniform Distribution
• A continuous random variable has a uniform
distribution if its values spread evenly over the
range of possibilities. The graph of a uniform
distribution results in a rectangular shape.

7
Uniform Distribution
• The uniform distribution is symmetric and
rectangular.

8
Requirements for a Probability Distribution
• where x assumes all possible
values.
• for every individual value of
x.

9
Density Curve
• A density curve is a graph of a continuous
probability distribution. It must satisfy the
following properties
• The total area under the curve must equal 1.
• Every point on the curve must have a vertical
height that is 0 or greater. (That is, the curve
cannot fall below the x-axis.)

10
Relationship Between Area Under the Curve and
Probability
• Because the total area under a density curve is
equal to 1, there is a correspondence between
area and probability.

11
Example
• Suppose that the continuous random variable X has
a uniform distribution over the interval from 0
to 5. Find the probability that a randomly
selected value of X is
• More than 4.
• Less than 2.
• Between 1 and 4.

12
Standard Normal Distribution
• 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.

13
Standard Normal Distribution
• The standard normal distribution

14
Probability
• A probability of falling in an interval is just
the area under the curve.

15
Probability

16
Probabilities and a Continuous Probability
Distribution
• For continuous numerical variables and any
particular numbers a and b,

17
Calculating Probabilities Given a z Score
• Table A-2 is designed only for the standard
normal distribution, which has a mean of 0 and a
standard deviation of 0.
• Table A-2 is on two pages, with one page for
negative z scores and the other page for positive
z scores.
• Each value in the body of the table is a
cumulative area from the left up to a vertical
boundary above the specific z score.

18
Calculating Probabilities Given a z Score
• When working with a graph, avoid confusion
between z scores and areas.
• z score Distance along the horizontal scale of
the standard normal distribution refer to the
leftmost column and top row of Table A-2.
• Area Region under the curve refer to the values
in the body of Table A-2.
• The part of the z score denoting hundredths is
found across the top row of Table A-2.

19
Example
• Find the area under the standard normal
distribution to the left of 1.5.
• Find the area under the standard normal
distribution to the right of -2.
• Find the area under the standard normal
distribution between -2 and 1.5.

20
Example
• Let z denote a random variable that has a
standard normal distribution. Determine each of
the following probabilities

21
Calculating a z Score Given a Probability
• Draw a bell-shaped curve and identify the region
under the curve that corresponds to the given
probability. If that region is not a cumulative
region from the left, work instead with a known
region that is a cumulative region from the left.
• Using the cumulative area from the left, locate
the closest probability in the body of Table A-2
and identify the corresponding z score.

22
Example
• Determine the z value that separates
• the smallest 10 of all the z values from the
others,
• the largest 5 of all the z values from the
others.

23
Applications of Normal Distributions
24
Standardizing Scores
• If we convert values to scores using
,then
procedures with all normal distributions are the
same as those for the standard normal
distribution.

25
Converting Values in a Nonstandard Normal
Distribution to z Scores
• Sketch a normal curve, label the mean and the
specific z values, then shade the region
representing the desired probability.
• For each relevant value x that is a boundary for
the shaded region, use the z Score formula to
convert that value to the equivalent z score.
• Refer to Table A-2 and use the z scores to find
the area of the shaded region. This area is the
desired probability.

26
Example
• The serum cholesterol levels of 17-year-olds
follow a normal distribution with a mean of 176
mg/dLi and a standard deviation of 30 mg/dLi. If
a 17-year-old is selected at random, what is the
probability he/she has a serum cholesterol level
• of 156 mg/dLi or less?
• of more than 216 mg/dLi?
• between 121 mg/dLi and 186 mg/dLi?

27
z Scores and Area
• Dont confuse z scores and areas.
• Choose the correct side of the graph.
• A z score must be negative whenever it is located
in the left half of the normal distribution.
• Areas (or probabilities) are positive or zero
values, but they are never negative.

28
Finding Values From Known Areas
• Sketch a normal distribution curve, enter the
given probability or percentage in the
appropriate region of the graph, and identify the
x value(s) being sought.
• Use Table A-2 to find the z score corresponding
to the cumulative left area bounded by x. Refer
to the body of Table A-2 to find the closest
area, then identify the corresponding z score.

29
Finding Values From Known Areas
• Using the formula
,enter the values for , ,
and the z score found in Step 2, then solve for
x.
• Refer to the sketch of the curve to verify that
the solution makes sense in the context of the
graph and in the context of the problem.

30
Example (continued)
• The serum cholesterol levels of 17-year-olds
follow a normal distribution with a mean of 176
mg/dLi and a standard deviation of 30 mg/dLi.
Find
• the 80th percentile.
• the 25th percentile.

31
Sampling Distributions and Estimators
32
Sampling Distribution of the Mean
• The sampling distribution of the mean is the
probability distribution of sample means, with
all samples having the same sample size n.

33
Example
• Suppose our population consists of the three
values 1, 2, and 5.
• Calculate the mean, median, range, variance and
standard deviation for the population.
• Find all possible samples of 2 values.
• Calculate the mean, median, range, variance and
standard deviation for each sample.
• Calculate the mean of the sample means, sample
medians, sample ranges, sample variances, and
sample standard deviation.
• Compare the results of d with the results of a.

34
Example (continued)
35
Sampling Variability
• The value of a statistic, such as the sample mean
, depends on the particular values included
in the sample, and it generally varies from
sample to sample. This variability of a statistic
is called sampling variability.

36
Sampling Distribution of the Proportion
• The sampling distribution of the proportion is
the probability distribution of sample
proportions, with all samples having the same
sample size n.

37
Properties of the Distribution of Sample
Proportions
• Sample proportions tend to target the value of
the population proportion.
• Under certain conditions, the distribution of
sample proportions approximates a normal
distribution.

38
Biased and Unbiased Estimators
• A sample statistic is an unbiased estimator of a
population parameter if it targets the
population parameter.
• A sample statistic is a biased estimator of a
population parameter if it does not target the
population parameter.

39
• Statistics that target population parameters
Mean, Variance, Proportion
• Statistics that do not target population
parameters Median, Range, Standard Deviation

40
The Central Limit Theorem
41
Example
• The serum cholesterol levels of 17-year-olds
follow a normal distribution with a mean of 176
mg/dLi and a standard deviation of 30 mg/dLi. If
a 17-year-old is selected at random, what is the
probability he/she has a serum cholesterol level
of 156 mg/dLi or less?

42
The Central Limit Theorem and the Sampling
Distribution of
• Given
• The random variable x has a distribution (which
may or may not be normal) with mean and
standard deviation .
• Simple random samples all of the same size n are
selected from the population. (The samples are
selected so that all possible samples of size n
have the same chance of being selected.)

43
The Central Limit Theorem and the Sampling
Distribution of
• Conclusions
• The distribution of sample means will, as the
sample increases, approach a normal distribution.
• The mean of all sample means is the population
mean . (That is, the normal distribution from
Conclusion 1 has mean .)
• The standard deviation of all sample means is
. (That is, the normal distribution from
Conclusion 1 has standard deviation .)

44
The Central Limit Theorem and the Sampling
Distribution of
• Practical Rules Commonly Used
• If the original population is not itself normally
distributed, here is a common guideline For
samples of size n greater than 30, the
distribution of the sample means can be
approximated reasonably well by a normal
distribution. (There are exceptions, such as
populations with very non-normal distributions
requiring samples sizes much larger than 30, but
such exceptions are relatively rare.) The
approximation gets better as the sample size n
becomes larger.
• If the original population is itself normally
distributed, then the sample means will be
normally distributed for any sample size n (not
just the values of n larger than 30).

45
Notation for the Sampling Distribution of
• If all possible random samples of size n are
selected from a population with mean and
standard deviation , the mean of the sample
means is denoted by , soAlso, the
standard deviation of the samples means is
denoted by , so is often called the
standard error of the mean.

46
Example
• The serum cholesterol levels of 17-year-olds
follow a normal distribution with a mean of 176
mg/dLi and a standard deviation of 30 mg/dLi. If
a random sample of ten 17-year-olds is selected,
what is the probability that the sample mean is
156 mg/dLi or less?

47
The Central Limit Theorem - Bottom Line
• As the sample size increases, the sampling
distribution of sample means approaches a normal
distribution.

48
Example
• If the mean and standard deviation of serum iron
values for healthy men are 120 and 15 micrograms
per 100 mL, respectively, what is the probability
that a random sample of 50 healthy men will yield
a sample mean between 115 and 125 micrograms per
100 mL?

49
Applying the Central Limit Theorem
• When working with an individual value from a
normally distributed population, use the methods
of Section 5.3. Use
• When working with a mean for same sample (or
group), be sure to use the value for
the standard deviation of the sample means. Use

50
Interpreting Results
• Rare Event RuleIf, under a given assumption, the
probability of a particular observed event is
exceptionally small, we conclude that the
assumption is probably not correct.

51
Correction for a Finite Population
• When sampling with replacement and the sample
size n is greater than 5 of the finite
population size N (that is, ),
adjust the standard deviation of the sample means
by multiplying it by the finite population
correction factor

52
Normal as Approximation to the Binomial
53
Recall
• A binomial probability distribution results from
a procedure that meets all the following
requirements
• The procedure has a fixed number of trials.
• The trials must be independent.
• Each trial must have all outcomes classified into
two categories.
• The probabilities must remain constant for each
trial.

54
Binomial Distributions p 0.5 n 3, n 4, n
5, n 6

55
Binomial Distributions p 0.5 n 10, n 20,
n 30, n 40

56
Binomial Distributions p 0.3 n 10, n 20,
n 30, n 40

57
Normal Distribution as Approximation to Binomial
Distribution
• When working with a binomial distribution, if
and , then the binomial random
variable has a probability distribution that can
be approximated by a normal distribution with the
mean and standard deviation given as

58
Definition
• When we use the normal distribution (which is a
continuous probability distribution) as an
approximation to the binomial distribution (which
is discrete), a continuity correction is made to
a discrete whole number x in the binomial
distribution by representing the single x value
by the interval from to
(that is, by adding and subtracting 0.5).

59
Example
• In a certain population of mussels (Mytilus
edulis), 80 of the individuals are infected with
an intestinal parasite. A marine biologist plans
to examine 100 randomly chosen mussels from the
population. Find the probability that at least 85
of the mussels will be infected.

60
Assessing Normality
61
Definition
• A normal quantile plot (or normal probability
plot) is a graph of the points (x, y) where each
x value is from the original set of sample data,
and each y value is the corresponding z score
that is a quantile value expected from the
standard normal distribution.

62
Procedure for Determining Whether Data Have a
Normal Distribution
• Histogram Construct a histogram. Reject
normality if the histogram departs dramatically
from a bell shape.
• Outliers Identify outliers. Reject normality if
there is more than one outlier present.
• Normal quantile plot If the histogram is
basically symmetric and there is at most one
outlier, construct a normal quantile plot.
Examine the normal quantile plot using these
criteria
• If the points do not lie close to a straight
line, or if the points exhibit some systematic
pattern that is not a straight-line pattern, then
the data appear to come from a population that
does not have a normal distribution.
• If the pattern of the points is reasonably close
to a straight line, then the data appear to come
from a population that has a normal distribution.

63
Example
• Recall our study of bears, the data for the
lengths of bears is given in Data Set 6 of
Appendix B. Determine whether the requirement of
a normal distribution is satisfied. Assume that
this requirement is loose in the sense that the
population distribution need not be exactly
normal, but it must be a distribution that is
basically symmetric with only one mode.

64
Example (continued)

65
Example (continued)

66
Example (continued)
• Using the weights of bears (given in Data Set 6
of Appendix B), determine whether the requirement
of a normal distribution is satisfied. Assume
that this requirement is loose in the sense that
the population distribution need not be exactly
normal, but it must be a distribution that is
basically symmetric with only one mode.

67
Example (continued)

68
Example (continued)

69
Data Transformations
• For data sets where the distribution is not
normal, we can transform the data so that the
modified values have a normal distribution.
Common transformations include