NORMAL PROBABILITY DISTRIBUTIONS

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NORMAL PROBABILITY DISTRIBUTIONS
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Overview
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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.

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The Normal Distribution

• The curve is bell-shaped and symmetric.

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The Standard Normal Distribution
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Requirements for a Probability Distribution

• where x assumes all possible
  values.
• for every individual value of
  x.

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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.)

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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.

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Probabilities and a Continuous Probability
Distribution

• For continuous numerical variables and any
  particular numbers a and b,

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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.

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Uniform Distribution

• The uniform distribution is symmetric and
  rectangular.

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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.

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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.

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Standard Normal Distribution

• The standard normal distribution

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Probability

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

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Probability

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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.

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Example

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

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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.

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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.

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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.

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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.

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Applications of Normal Distributions
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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.

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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

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Example (continued)

• less than 72 inches?
• more than 64 inches?
• between 64 inches and 72 inches?

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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.

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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.

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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.

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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.

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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.

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Sampling Distributions and Estimators
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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.)

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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.)

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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.

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Example (continued)
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Example (continued)
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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.

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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.

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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.

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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.

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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

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The Central Limit Theorem
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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?

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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.)

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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 .)

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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).

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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.

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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?

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The Central Limit Theorem The Bottom Line

• As the sample size increases, the sampling
  distribution of sample means approaches a normal
  distribution.

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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?

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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

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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.

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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

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Normal as Approximation to the Binomial
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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.

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

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

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

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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
  .

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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).

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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.

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Assessing Normality
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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.

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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.

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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.

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Example (continued)

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Example (continued)

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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.

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Example (continued)

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Example (continued)

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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