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.

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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 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.

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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.

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.

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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.

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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.

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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).

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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.

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

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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.

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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

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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

• 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.

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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.

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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)

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.

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Example (continued)

68
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