NORMAL PROBABILITY DISTRIBUTIONS - PowerPoint PPT Presentation

Loading...

PPT – NORMAL PROBABILITY DISTRIBUTIONS PowerPoint presentation | free to view - id: 119a3b-N2JjY



Loading


The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
About This Presentation
Title:

NORMAL PROBABILITY DISTRIBUTIONS

Description:

Normal Quantile Plot ... Normal quantile plot: If the histogram is basically symmetric and there is at ... Examine the normal quantile plot using these criteria: ... – PowerPoint PPT presentation

Number of Views:618
Avg rating:3.0/5.0
Slides: 73
Provided by: wild6
Category:

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

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

7
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.)

8
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.

9
Probabilities and a Continuous Probability
Distribution
  • For continuous numerical variables and any
    particular numbers a and b,

10
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.

11
Uniform Distribution
  • The uniform distribution is symmetric and
    rectangular.

12
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 3,
  • Less than 1,
  • Between 2 and 4.

13
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.

14
Standard Normal Distribution
  • The standard normal distribution

15
Probability
  • A probability of falling in an interval is just
    the area under the curve.

16
Probability

17
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.

18
Example
  • Let z denote a random variable that has a
    standard normal distribution. Determine each of
    the following probabilities

19
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.

20
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.

21
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.

22
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.

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
Example
  • Suppose the heights of adult males in the
    population have a normal distribution with a mean
    of 70 inches and a standard deviation of 2.8
    inches. An adult male is selected at random, what
    is the probability that his height is

26
Example (continued)
  • less than 72 inches?
  • more than 64 inches?
  • between 64 inches and 72 inches?

27
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.

28
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.

29
Example (continued)
  • The heights of adult males in the population have
    a normal distribution with a mean of 70 inches
    and a standard deviation of 2.8 inches. Find
  • the 80th percentile.
  • the 25th percentile.

30
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.

31
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.

32
Sampling Distributions and Estimators
33
Sampling Distribution of a Statistic
  • The sampling distribution of a statistic (such as
    a sample proportion or sample mean) is the
    distribution of all values of the statistic when
    all possible samples of the same size n are taken
    from the same population. (The sampling
    distribution of a statistic is typically
    represented as a probability distribution in the
    format of a table, probability histogram, or
    formula.)

34
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. (The
    sampling distribution of the mean is typically
    represented as a probability distribution in the
    format of a table, probability histogram, or
    formula.)

35
Example
  • Suppose our population consists of the three
    values 1, 2, and 5.
  • Calculate the mean, 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.

36
Example (continued)
37
Example (continued)
38
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.

39
Sampling Distribution of the Proportion
  • The sampling distribution of the proportion is
    the distribution of sample proportions, with all
    samples having the same sample size n taken from
    the same population.

40
Properties of the Sampling Distribution of the
Proportion
  • Sample proportions tend to target the value of
    the population proportion.
  • Under certain conditions, the distribution of
    sample proportions approximates a normal
    distribution.

41
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.

42
Which Statistics Make Good Estimators of
Parameters?
  • Statistics that target population parameters
    Mean, Variance, Proportion
  • Statistics that do not target population
    parameters Median, Range, Standard Deviation

43
The Central Limit Theorem
44
Example
  • Suppose the heights of adult males in the
    population have a normal distribution with a mean
    of 70 inches and a standard deviation of 2.8
    inches. An adult male is selected at random, what
    is the probability that his height is less than
    68 inches?

45
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.)

46
The Central Limit Theorem and the Sampling
Distribution of
  • Conclusions
  • The distribution of sample means will, as the
    sample size 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 .)

47
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).

48
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.

49
Example
  • Suppose the heights of adult males in the
    population have a normal distribution with a mean
    of 70 inches and a standard deviation of 2.8
    inches. If a random sample of ten adult males is
    selected, what is the probability that the sample
    mean is less than 68 inches?

50
The Central Limit Theorem The Bottom Line
  • As the sample size increases, the sampling
    distribution of sample means approaches a normal
    distribution.

51
Example
  • According to the Energy Information
    Administration, the mean household size in the
    United States in 1997 was 2.6 people, with a
    standard deviation of 1.5 people. What is the
    probability that a random sample of 100
    households results in a sample mean household
    size of 2.4 or less?

52
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

53
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.

54
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

55
Normal as Approximation to the Binomial
56
The Binomial Distribution Recap
  • 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.

57
Binomial Distributions p 0.5 n 3, n 4, n
5, n 6

58
Binomial Distributions p 0.5 n 10, n 20,
n 30, n 40

59
Binomial Distributions p 0.3 n 10, n 20,
n 30, n 40

60
Normal Distribution as Approximation to Binomial
Distribution
  • If a binomial probability distribution satisfies
    the requirements and ,
    then that binomial probability distribution that
    can be approximated by a normal distribution with
    mean and standard deviation
    , and with discrete whole number x adjusted
    with a continuity correction, so that x is
    represented by the interval from to
    .

61
Continuity Corrections
  • 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).

62
Example
  • According to Information Please almanac, 80 of
    adult smokers started smoking before they were 18
    years old. Suppose 100 smokers 18 years old or
    older are randomly selected. What is the
    probability that that
  • Fewer than 70 of them started smoking before they
    were 18 years old.
  • Exactly 80 of them started smoking before they
    were 18 years old.

63
Assessing Normality
64
Normal Quantile Plot
  • 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.

65
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.

66
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.

67
Example (continued)

68
Example (continued)

69
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.

70
Example (continued)

71
Example (continued)

72
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
About PowerShow.com