Statistics Test

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Lecture Slides

Elementary Statistics Eleventh Edition

and the Triola Statistics Series

by Mario F. Triola

Chapter 6

Normal Probability Distributions

6-1 Review and Preview

6-2 The Standard Normal Distribution

6-3 Applications of Normal Distributions

6-4 Sampling Distributions and Estimators

6-5 The Central Limit Theorem

Section 6-1

Review and Preview

❖ Chapter 2: Distribution of data

❖ Chapter 3: Measures of data sets,

including measures of center and

variation

❖ Chapter 4: Principles of probability

❖ Chapter 5: Discrete probability

distributions

Review

Chapter focus is on:

❖ Continuous random variables

❖ Normal distributions

Preview

Figure 6-1

Formula 6-1

f x( )= e − 1

x−





 



 2

Distribution determined

by fixed values of mean

and standard deviation

Section 6-2

The Standard Normal

Distribution

Key Concept

This section presents the standard normal

distribution which has three properties:

1. It’s graph is bell-shaped.

2. It’s mean is equal to 0 ( = 0).

3. It’s standard deviation is equal to 1 ( = 1).

Develop the skill to find areas (or probabilities

or relative frequencies) corresponding to

various regions under the graph of the

standard normal distribution. Find z-scores that

correspond to area under the graph.

Uniform Distribution

A continuous random variable has a

uniform distribution if its values are

spread evenly over the range of

probabilities. The graph of a uniform

distribution results in a rectangular shape.

A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:

Density Curve

1. The total area under the curve must equal 1.

2. 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.)

Because the total area under the

density curve is equal to 1,

there is a correspondence

between area and probability.

Area and Probability

Using Area to Find Probability

Given the uniform distribution illustrated, find the probability that a randomly selected voltage level is greater than 124.5 volts.

Shaded area

represents

voltage levels

greater than

124.5 volts.

Correspondence

between area

and probability:

0.25.

Standard Normal Distribution

The standard normal distribution is a

normal probability distribution with  = 0

and  = 1. The total area under its density

curve is equal to 1.

Finding Probabilities When

Given z-scores

❖ Table A-2 (in Appendix A)

❖ Formulas and Tables insert card

❖ Find areas for many different regions

Finding Probabilities –

Other Methods

❖ STATDISK

❖ Minitab

❖ Excel

❖ TI-83/84 Plus

Methods for Finding Normal

Distribution Areas

Methods for Finding Normal

Distribution Areas

Table A-2

1. It is designed only for the standard normal distribution, which has a mean of 0 and a standard deviation of 1.

2. It is on two pages, with one page for negative z-scores and the other page for positive z-scores.

3. Each value in the body of the table is a cumulative area from the left up to a vertical boundary above a specific z-score.

Using Table A-2

4. When working with a graph, avoid confusion

between z-scores and areas.

z Score

Distance along 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.

5. The part of the z-score denoting hundredths

is found across the top.

Using Table A-2

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0ºC at the freezing point of water. Tests on a large sample of these instruments reveal that at the freezing point of water, some thermometers give readings below 0º (denoted by negative numbers) and some give readings above 0º (denoted by positive numbers). Assume that the mean reading is 0ºC and the standard deviation of the readings is 1.00ºC. Also assume that the readings are normally distributed. If one thermometer is randomly selected, find the probability that, at the freezing point of water, the reading is less than 1.27º.

Example - Thermometers

P(z < 1.27) =

Example - (Continued)

Look at Table A-2

P (z < 1.27) = 0.8980

Example - cont

The probability of randomly selecting a thermometer with a reading less than 1.27º is 0.8980.

P (z < 1.27) = 0.8980

Example - cont

Or 89.80% will have readings below 1.27º.

P (z < 1.27) = 0.8980

Example - cont

If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water, and if one thermometer is randomly selected, find the probability that it reads (at the freezing point of water) above –1.23 degrees.

Probability of randomly selecting a thermometer with a reading above –1.23º is 0.8907.

P (z > –1.23) = 0.8907

Example - Thermometers Again

P (z > –1.23) = 0.8907

89.07% of the thermometers have readings above –1.23 degrees.

Example - cont

A thermometer is randomly selected. Find the probability that it reads (at the freezing point of water) between –2.00 and 1.50 degrees.

P (z < –2.00) = 0.0228 P (z < 1.50) = 0.9332 P (–2.00 < z < 1.50) = 0.9332 – 0.0228 = 0.9104

The probability that the chosen thermometer has a

reading between – 2.00 and 1.50 degrees is 0.9104.

Example - Thermometers III

If many thermometers are selected and tested at the freezing point of water, then 91.04% of them will read between –2.00 and 1.50 degrees.

P (z < –2.00) = 0.0228 P (z < 1.50) = 0.9332 P (–2.00 < z < 1.50) = 0.9332 – 0.0228 = 0.9104

A thermometer is randomly selected. Find the probability that it reads (at the freezing point of water) between –2.00 and 1.50 degrees.

Example - cont

P(a < z < b) denotes the probability that the z score is between a and b.

P(z > a)

denotes the probability that the z score is greater than a.

P(z < a) denotes the probability that the z score is less than a.

Notation

Finding a z Score When Given a

Probability Using Table A-2

1. 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.

2. Using the cumulative area from the left, locate the closest probability in the body of Table A-2 and identify the corresponding z score.

Finding z Scores When Given Probabilities

5% or 0.05

(z score will be positive)

Finding the 95th Percentile

Finding z Scores When Given Probabilities - cont

Finding the 95th Percentile

1.645

5% or 0.05

(z score will be positive)

Finding the Bottom 2.5% and Upper 2.5%

(One z score will be negative and the other positive)

Finding z Scores When Given Probabilities - cont

Finding the Bottom 2.5% and Upper 2.5%

(One z score will be negative and the other positive)

Finding z Scores When Given Probabilities - cont

Finding the Bottom 2.5% and Upper 2.5%

(One z score will be negative and the other positive)

Finding z Scores When Given Probabilities - cont

Recap

In this section we have discussed:

❖ Density curves.

❖ Relationship between area and probability.

❖ Standard normal distribution.

❖ Using Table A-2.

Section 6-3

Applications of Normal

Distributions

Key Concept

This section presents methods for working

with normal distributions that are not standard.

That is, the mean is not 0 or the standard

deviation is not 1, or both.

The key concept is that we can use a simple

conversion that allows us to standardize any

normal distribution so that the same methods

of the previous section can be used.

Conversion Formula

x – µ z =

Round z scores to 2 decimal places

Converting to a Standard

Normal Distribution

x – 

 z =

In the Chapter Problem, we noted that the safe

load for a water taxi was found to be 3500

pounds. We also noted that the mean weight of a

passenger was assumed to be 140 pounds.

Assume the worst case that all passengers are

men. Assume also that the weights of the men

are normally distributed with a mean of 172

pounds and standard deviation of 29 pounds. If

one man is randomly selected, what is the

probability he weighs less than 174 pounds?

Example – Weights of

Water Taxi Passengers

Example - cont

z = 174 – 172

29 = 0.07

 = 29

 = 172

Example - cont

P ( x < 174 lb.) = P(z < 0.07)

= 0.5279

 = 29

 = 172

1. Don’t confuse z scores and areas. z scores are

distances along the horizontal scale, but areas

are regions under the normal curve. Table A-2

lists z scores in the left column and across the top

row, but areas are found in the body of the table.

2. Choose the correct (right/left) side of the graph.

3. A z score must be negative whenever it is located

in the left half of the normal distribution.

4. Areas (or probabilities) are positive or zero values,

but they are never negative.

Helpful Hints

Procedure for Finding Values

Using Table A-2 and Formula 6-2 1. 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.

2. 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.

3. Using Formula 6-2, enter the values for µ, , and the z score found in step 2, then solve for x.

x = µ + (z • ) (Another form of Formula 6-2)

(If z is located to the left of the mean, be sure that it is a negative number.)

4. Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and the context of the problem.

Example – Lightest and Heaviest

Use the data from the previous example to determine

what weight separates the lightest 99.5% from the

heaviest 0.5%?

x =  + (z ● )

x = 172 + (2.575 • 29)

x = 246.675 (247 rounded)

Example –

Lightest and Heaviest - cont

The weight of 247 pounds separates the

lightest 99.5% from the heaviest 0.5%

Example –

Lightest and Heaviest - cont

Recap

In this section we have discussed:

❖ Non-standard normal distribution.

❖ Converting to a standard normal distribution.

❖ Procedures for finding values using Table A-2

and Formula 6-2.

Section 6-4

Sampling Distributions

and Estimators

Key Concept

The main objective of this section is to

understand the concept of a sampling

distribution of a statistic, which is the

distribution of all values of that statistic

when all possible samples of the same size

are taken from the same population.

We will also see that some statistics are

better than others for estimating population

parameters.

Definition

The sampling distribution of a statistic (such

as the sample mean or sample proportion) 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.)

Definition

The sampling distribution of the mean is

the distribution of sample means, with all

samples having the same sample size n

taken from the same population. (The

sampling distribution of the mean is

typically represented as a probability

distribution in the format of a table,

probability histogram, or formula.)

Properties

❖ Sample means target the value of the

population mean. (That is, the mean of the

sample means is the population mean. The

expected value of the sample mean is equal

to the population mean.)

❖ The distribution of the sample means tends

to be a normal distribution.

Definition

The sampling distribution of the variance is the

distribution of sample variances, with all

samples having the same sample size n taken

from the same population. (The sampling

distribution of the variance is typically

represented as a probability distribution in the

format of a table, probability histogram, or

formula.)

Properties

❖ Sample variances target the value of the

population variance. (That is, the mean of

the sample variances is the population

variance. The expected value of the sample

variance is equal to the population variance.)

❖ The distribution of the sample variances

tends to be a distribution skewed to the

right.

Definition

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.

Definition

We need to distinguish between a

population proportion p and some sample

proportion:

p = population proportion

= sample proportion p̂

Properties

❖ Sample proportions target the value of the

population proportion. (That is, the mean of

the sample proportions is the population

proportion. The expected value of the

sample proportion is equal to the population

proportion.)

❖ The distribution of the sample proportion

tends to be a normal distribution.

Unbiased Estimators

Sample means, variances and

proportions are unbiased estimators.

That is they target the population

parameter.

These statistics are better in estimating

the population parameter.

Biased Estimators

Sample medians, ranges and standard

deviations are biased estimators.

That is they do NOT target the

population parameter.

Note: the bias with the standard

deviation is relatively small in large

samples so s is often used to estimate.

Example - Sampling Distributions

Consider repeating this process: Roll a die 5

times, find the mean , variance s2, and the

proportion of odd numbers of the results.

What do we know about the behavior of all

sample means that are generated as this

process continues indefinitely?

Example - Sampling Distributions

All outcomes are equally likely so the

population mean is 3.5; the mean of the

10,000 trials is 3.49. If continued indefinitely,

the sample mean will be 3.5. Also, notice the

distribution is “normal.”

Specific results from 10,000 trials

Example - Sampling Distributions

All outcomes are equally likely so the

population variance is 2.9; the mean of the

10,000 trials is 2.88. If continued indefinitely,

the sample variance will be 2.9. Also, notice

the distribution is “skewed to the right.”

Specific results from 10,000 trials

Example - Sampling Distributions

All outcomes are equally likely so the

population proportion of odd numbers is 0.50;

the proportion of the 10,000 trials is 0.50. If

continued indefinitely, the mean of sample

proportions will be 0.50. Also, notice the

distribution is “approximately normal.”

Specific results from 10,000 trials

Why Sample with Replacement?

Sampling without replacement would have the very

practical advantage of avoiding wasteful duplication

whenever the same item is selected more than once.

However, we are interested in sampling with

replacement for these two reasons:

1. When selecting a relatively small sample form a

large population, it makes no significant

difference whether we sample with replacement

or without replacement.

2. Sampling with replacement results in

independent events that are unaffected by

previous outcomes, and independent events are

easier to analyze and result in simpler

calculations and formulas.

Caution

Many methods of statistics require a simple

random sample. Some samples, such as

voluntary response samples or convenience

samples, could easily result in very wrong

results.

Recap

In this section we have discussed:

❖ Sampling distribution of a statistic.

❖ Sampling distribution of the mean.

❖ Sampling distribution of the variance.

❖ Sampling distribution of the proportion.

❖ Estimators.

Section 6-5

The Central Limit

Theorem

Key Concept

The Central Limit Theorem tells us that for a

population with any distribution, the

distribution of the sample means approaches

a normal distribution as the sample size

increases.

The procedure in this section form the

foundation for estimating population

parameters and hypothesis testing.

Central Limit Theorem

1. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation .

2. Simple random samples all of size n are selected from the population. (The samples are selected so that all possible samples of the same size n have the same chance of being selected.)

Given:

1. The distribution of sample x will, as the

sample size increases, approach a normal

distribution.

2. The mean of the sample means is the

population mean µ.

3. The standard deviation of all sample means

Conclusions:

Central Limit Theorem – cont.

 n.

Practical Rules Commonly Used

1. For samples of size n larger than 30, the

distribution of the sample means can be

approximated reasonably well by a normal

distribution. The approximation gets closer

to a normal distribution as the sample size n

becomes larger.

2. If the original population is normally

distributed, then for any sample size n, the

sample means will be normally distributed

(not just the values of n larger than 30).

Notation

the mean of the sample means

the standard deviation of sample mean

(often called the standard error of the mean)

µx = µ

nx = 

Example - Normal Distribution

As we proceed

from n = 1 to

n = 50, we see

that the

distribution of

sample means

is approaching

the shape of a

normal

distribution.

Example - Uniform Distribution

As we proceed

from n = 1 to

n = 50, we see

that the

distribution of

sample means

is approaching

the shape of a

normal

distribution.

Example - U-Shaped Distribution

As we proceed

from n = 1 to

n = 50, we see

that the

distribution of

sample means

is approaching

the shape of a

normal

distribution.

As the sample size increases, the

sampling distribution of sample

means approaches a normal

distribution.

Important Point

Use the Chapter Problem. Assume the population of weights of men is normally distributed with a mean of 172 lb and a standard deviation of 29 lb.

Example – Water Taxi Safety

a) Find the probability that if an individual man is randomly selected, his weight is greater than 175 lb.

b) b) Find the probability that 20 randomly selected men will have a mean weight that is greater than 175 lb (so that their total weight exceeds the safe capacity of 3500 pounds).

z = 175 – 172 = 0.10 29

a) Find the probability that if an individual man is randomly selected, his weight is greater than 175 lb.

Example – cont

b) Find the probability that 20 randomly selected men will have a mean weight that is greater than 175 lb (so that their total weight exceeds the safe capacity of 3500 pounds).

Example – cont

z = 175 – 172 = 0.46 29

b) Find the probability that 20 randomly selected men

will have a mean weight that is greater than 175 lb (so

that their total weight exceeds the safe capacity of

3500 pounds).

It is much easier for an individual to deviate from the mean than it is for a group of 20 to deviate from the mean.

a) Find the probability that if an individual man is

randomly selected, his weight is greater than 175 lb.

Example - cont

P(x > 175) = 0.4602

P(x > 175) = 0.3228

Interpretation of Results

Given that the safe capacity of the water taxi

is 3500 pounds, there is a fairly good chance

(with probability 0.3228) that it will be

overloaded with 20 randomly selected men.

Correction for a Finite Population

N – n x

=  n N – 1

finite population correction factor

When sampling without replacement and the sample

size n is greater than 5% of the finite population of

size N (that is, n > 0.05N ), adjust the standard

deviation of sample means by multiplying it by the

finite population correction factor:

Recap

In this section we have discussed:

❖ Central limit theorem.

❖ Practical rules.

❖ Effects of sample sizes.

❖ Correction for a finite population.

Section 6-7

Assessing Normality

Key Concept

This section presents criteria for determining

whether the requirement of a normal

distribution is satisfied.

The criteria involve visual inspection of a

histogram to see if it is roughly bell shaped,

identifying any outliers, and constructing a

graph called a normal quantile plot.

Definition

A normal quantile plot (or normal

probability plot) is a graph of 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

It Is Reasonable to Assume that

Sample Data are From a Normally

Distributed Population 1. Histogram: Construct a histogram. Reject

normality if the histogram departs dramatically from a bell shape.

2. Outliers: Identify outliers. Reject normality if there is more than one outlier present.

3. Normal Quantile Plot: If the histogram is basically symmetric and there is at most one outlier, use technology to generate a normal quantile plot.

Procedure for Determining Whether

It Is Reasonable to Assume that

Sample Data are From a Normally

Distributed Population 3. Continued

Use the following criteria to determine whether or not the distribution is normal.

Normal Distribution: The population distribution is normal if the pattern of the points is reasonably close to a straight line and the points do not show some systematic pattern that is not a straight-line pattern.

Procedure for Determining Whether

It Is Reasonable to Assume that

Sample Data are From a Normally

Distributed Population 3. Continued

Not a Normal Distribution: The population distribution is

not normal if either or both of these two conditions

applies:

❖ The points do not lie reasonably close to a straight

line.

❖ The points show some systematic pattern that is not a

straight-line pattern.

Example

Normal: Histogram of IQ scores is close to being bell-

shaped, suggests that the IQ scores are from a normal

distribution. The normal quantile plot shows points that

are reasonably close to a straight-line pattern. It is safe to

assume that these IQ scores are from a normally

distributed population.

Example

Uniform: Histogram of data having a uniform distribution.

The corresponding normal quantile plot suggests that the

points are not normally distributed because the points

show a systematic pattern that is not a straight-line

pattern. These sample values are not from a population

having a normal distribution.

Example

Skewed: Histogram of the amounts of rainfall in Boston for

every Monday during one year. The shape of the histogram

is skewed, not bell-shaped. The corresponding normal

quantile plot shows points that are not at all close to a

straight-line pattern. These rainfall amounts are not from a

population having a normal distribution.

Manual Construction of a

Normal Quantile Plot Step 1. First sort the data by arranging the values in

order from lowest to highest.

Step 2. With a sample of size n, each value represents a

proportion of 1/n of the sample. Using the known

sample size n, identify the areas of 1/2n, 3/2n,

and so on. These are the cumulative areas to the

left of the corresponding sample values.

Step 3. Use the standard normal distribution (Table A-2

or software or a calculator) to find the z scores

corresponding to the cumulative left areas found

in Step 2. (These are the z scores that are

expected from a normally distributed sample.)

Manual Construction of a

Normal Quantile Plot

Step 4. Match the original sorted data values with their

corresponding z scores found in Step 3, then

plot the points (x, y), where each x is an original

sample value and y is the corresponding z score.

Step 5. Examine the normal quantile plot and determine

whether or not the distribution is normal.

Ryan-Joiner Test

The Ryan-Joiner test is one of several formal

tests of normality, each having their own

advantages and disadvantages. STATDISK has a

feature of Normality Assessment that displays a

histogram, normal quantile plot, the number of

potential outliers, and results from the Ryan-

Joiner test. Information about the Ryan-Joiner

test is readily available on the Internet.

Data Transformations

Many data sets have a distribution that is not

normal, but we can transform the data so that

the modified values have a normal distribution.

One common transformation is to replace each

value of x with log (x + 1). If the distribution of

the log (x + 1) values is a normal distribution,

the distribution of the x values is referred to as

a lognormal distribution.

Other Data Transformations

In addition to replacing each x value with the

log (x + 1), there are other transformations,

such as replacing each x value with , or 1/x,

or x2. In addition to getting a required normal

distribution when the original data values are

not normally distributed, such transformations

can be used to correct other deficiencies, such

as a requirement (found in later chapters) that

different data sets have the same variance.

Recap

In this section we have discussed:

❖ Normal quantile plot.

❖ Procedure to determine if data have a

normal distribution.