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NORMAL PROBABILITY DISTRIBUTIONS

Overview

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.

The Normal Distribution

- The curve is bell-shaped and symmetric.

The Standard Normal Distribution

Requirements for a Probability Distribution

- where x assumes all possible

values. - for every individual value of

x.

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.)

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.

Probabilities and a Continuous Probability

Distribution

- For continuous numerical variables and any

particular numbers a and b,

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.

Uniform Distribution

- The uniform distribution is symmetric and

rectangular.

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.

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.

Standard Normal Distribution

- The standard normal distribution

Probability

- A probability of falling in an interval is just

the area under the curve.

Probability

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.

Example

- Let z denote a random variable that has a

standard normal distribution. Determine each of

the following probabilities

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.

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.

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.

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.

Applications of Normal Distributions

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.

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

Example (continued)

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

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.

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.

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.

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.

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.

Sampling Distributions and Estimators

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.)

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.)

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.

Example (continued)

Example (continued)

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.

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.

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.

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.

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

The Central Limit Theorem

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?

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.)

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 .)

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).

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.

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?

The Central Limit Theorem The Bottom Line

- As the sample size increases, the sampling

distribution of sample means approaches a normal

distribution.

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?

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

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.

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

Normal as Approximation to the Binomial

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.

Binomial Distributions p 0.5 n 3, n 4, n

5, n 6

Binomial Distributions p 0.5 n 10, n 20,

n 30, n 40

Binomial Distributions p 0.3 n 10, n 20,

n 30, n 40

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

.

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).

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.

Assessing Normality

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.

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.

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.

Example (continued)

Example (continued)

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.

Example (continued)

Example (continued)

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