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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
Estimators Good and Bad
  • 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
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