Name _____________________

1

Focus Points:

Graph a normal

curve and summarize

its important

properties

Apply the empirical

rule to solve real-

world problems

Chapter 7 Normal Curves & Sampling Distributions

Section 7.1 Graphs of Normal Probability Distributions

Objective: In this lesson you learned how to graph a normal curve and apply the empirical rule to

solve real-world problems.

What is the graph of a normal distribution called?

What is another name for a normal curve?

What does the parameter 𝜎 (standard deviation) control?

If 𝜎 is large what happens to the curve? If 𝜎 is small?

Important Properties of a Normal Curve

1.

2.

3.

4.

5.

Important Vocabulary

Normal Curve Empirical Rule

2

Why is the area under a normal curve important?

Empirical Rule

How does the application of the empirical rule vary from Chebyshev’s Theorem?

What does a Normal Distribution tell us?

Figure 7-1

3

Section 7.1 Example – Graphs of Normal Probability Distributions

( 1 ) The yearly wheat yield per acre on a particular farm is normally distributed with mean 𝜇 = 35 bushels

and standard deviation 𝜎 = 8 bushels.

a. Shade the area under the curve, in the

figure to the right, that represents the

probability that an acre will yield between

19 and 35 bushels.

b. Is the area the same as the area between 𝜇 − 2𝜎 and 𝜇?

c. Use Figure 7-1 (from your notes) to find the percentage of area over the interval between 19

and 35.

d. What is the probability that the yield will be between 19 and 35 bushels per acre?

4

Focus Points:

Given 𝜇 and 𝜎,

convert raw scores to

𝑧-scores

Given 𝜇 and 𝜎,

convert 𝑧-scores to

raw scores

Focus Point:

Graph the standard

normal distribution

Section 7.2 Standard Units and Areas Under the Standard Normal Distribution

Objective: In this lesson you learned to convert between raw scores and 𝑧-scores; graph standard

normal distributions; and find areas under the standard normal curve.

I. 𝒛-Scores and Raw Scores

In what ways do normal distributions vary from each other?

In standard units, what is the mean of the original distribution?

A 𝑥-value that is above the original mean is what?

A 𝑥-value that is below the original mean is what?

What is a raw score? How is it calculated?

II. Standard Normal Distribution

Important Vocabulary

𝑧-score Standard Score Raw Score Standard Normal Distribution

The 𝒛-value

The standard normal distribution

5

Focus Point:

Find areas under the

Standard Normal

Curve

What does the Standard Normal Distribution tell us?

III. Areas Under the Standard Normal Curve

What is the advantage to converting a normal distribution to a

standard distribution?

Why are the area(s) under the curve important?

IV. Using a Standard Normal Distribution Table

How to use a Left-Tail style Standard Normal Distribution Table

1.

2.

3.

6

Section 7.2 Examples – Standard Units and Areas Under the Standard Normal

Distribution

( 1 ) Rod figures that it takes an average (mean) of 17 minutes with a standard deviation of 3 minutes to

drive from home, park the car, and walk to an early morning class.

a. One day it took Rod 21 minutes to get to class. How many standard deviations from the average

is that? Is the 𝑧-value positive or negative? Explain why the 𝑧-value should be either positive or

negative.

b. What commuting time corresponds to a standard score of 𝑧 = −2.5? Could Rod count on

making it to class in this amount of time or less?

( 2 ) Table 3, Areas of a Standard Normal Distribution, is located in the Appendix as well as in the endpapers

of the text. Spend a little time studying the table, and then answer these questions.

a. As 𝑧-values increase, do the areas to the left of 𝑧 increase?

b. If a 𝑧 value is negative, is the area to the left of 𝑧 less than 0.5000?

c. If a 𝑧 value is positive, is the area to the left of 𝑧 greater than 0.5000?

( 3 ) Let 𝑧 be a random variable with a standard normal distribution.

a. 𝑃(𝑧 ≥ 1.15) refers to the probability that 𝑧-values lie to the right of 1.15. Draw and shade the

corresponding area under the standard normal curve and find 𝑃(𝑧 ≥ 1.15).

b. Find 𝑃(−1.78 ≤ 𝑧 ≤ 0.35). First, sketch the area under the standard normal curve

corresponding to 𝑃(−1.78 ≤ 𝑧 ≤ 0.35).

7

Focus Point:

Compute the

probability of

“standardized

events”

Focus Point:

Find a 𝑧-score from a

given normal

probability (inverse

normal)

Section 7.3 Areas Under Any Normal Curve

Objective: In this lesson you learned to compute probabilities of “standardized events” and find a

𝑧-score from a given normal probability.

I. Normal Distribution Areas

II. Inverse Normal Distribution

What is inverse normal probability distribution?

III. Checking for Normality

1. Histogram:

2. Outliers:

3. Skewness:

How to work with Normal Distributions

8

Section 7.3 Examples – Areas Under Any Normal Curve

( 1 ) The life span of a rechargeable battery is the time before the battery must be replaced because it no

longer holds a charge. One tablet computer model has a battery with a life span that is normally

distributed with a mean of 2.3 years and a standard deviation of 0.4 year. What is the probability that

the battery will have to be replaced during the guarantee period of 2 years?

a. Let 𝑥 represent the battery life span. The statement that the battery needs to be replaced during

the 2-year guarantee period means the life span is less than 2 years, or 𝑥 ≤ 2. Convert this

statement to a statement about 𝑧.

b. Indicate the area to be found, in the figure below, by shading. Does this area correspond to the

probability that 𝑧 ≤ −0.75?

c. Use the Standard Normal Distribution table to find 𝑃(𝑧 ≤ −0.75).

d. What is the probability that the battery will fail before the end of the guarantee period?

( 2 ) Find the 𝑧-value such that 3% of the area under the standard normal curve lies to the right of 𝑧.

a. Draw a sketch of the standard normal distribution showing the described area.

b. Find the area to the left of 𝑧.

c. Look up the area in the Standard Normal Distribution table and find the corresponding 𝑧-value.

9

d. Suppose the time to complete a test is normally distributed with 𝜇 = 40 minutes and 𝜎 = 5

minutes. After how many minutes can we expect all but about 3% of the tests to be completed?

e. Use the Standard Normal Distribution table to find a 𝑧-value such that 3% of the area under the

standard normal curve lies to the left of 𝑧.

10

Focus Points:

Review such

commonly used

terms as random

sample, relative

frequency,

parameter, statistic,

and sampling

distribution

From raw data,

construct a relative

frequency

distribution for �̅�

values and compare

the result to a

theoretical sampling

distribution

Section 7.4 Sampling Distributions

Objective: In this lesson you learned to construct a relative frequency distribution for �̅� values from

raw data.

What is a population, in terms of statistics?

What is a sample?

What is a statistic?

What is a parameter?

Important Vocabulary

Statistic Parameter Sampling Distribution

11

Types of Inferences

1. Estimation:

2. Testing:

3. Regression:

What does a sampling distribution tell us?

A sampling distribution

12

Section 7.4 Examples – Sampling Distributions

( 1 ) What is a population parameter? Give an example.

( 2 ) What is a sample statistic? Give an example.

( 3 ) What is a sampling distribution?

13

Focus Point:

For a normal

distribution, use 𝜇

and 𝜎 to construct

the theoretical

sampling distribution

for the statistic �̅�

Section 7.5 The Central Limit Theorem

Objective: In this lesson you learned to construct theoretical sampling distributions for the statistic

�̅�; use sample estimates to construct a good approximate sampling distribution for the

statistic �̅�; the underlying meaning of the central limit theorem.

I. The �̅� Distribution, Given 𝒙 is Normal

What is the standard deviation of a statistic referred to as?

Important Vocabulary

Standard Error Central Limit Theorem Large Sample

Theorem 7.1 – For a Normal Probability Distribution

The standard error

14

Focus Point:

For large samples,

use sample estimates

to construct a good

approximate

sampling distribution

for the statistic �̅�

II. The �̅� Distribution, Given 𝒙 Has an Distribution

How large should a sample be for the �̅� distribution to appear normal?

What does the Central Limit Theorem tell us?

Theorem 7.2 – The Central Limit Theorem

Using the Central Limit Theorem to convert the �̅� Distribution

15

How to find Probabilities Regarding 𝑥

1.

2.

3.

4.

16

Section 7.5 Examples – The Central Limit Theorem

( 1 ) Using the Central Limit Theorem

a. Suppose 𝑥 has a normal distribution with mean 𝜇 = 18 and standard deviation 𝜎 = 13. If you

draw random samples of size 5 from the 𝑥 distribution and �̅� represents the sample mean, what

can you say about the �̅� distribution? How could you standardize the �̅� distribution?

b. Suppose you know that the 𝑥 distribution has mean 𝜇 = 75 and standard deviation 𝜎 = 12, but

you have no information as to whether or not the 𝑥 distribution is normal. If you draw samples

of size 30 from the 𝑥 distribution and �̅� represents the sample mean, what can you say about

the �̅� distribution? How could you standardize the �̅� distribution?

c. Suppose you did not know that 𝑥 had a normal distribution. Would you be justified in saying that

the �̅� distribution is approximately normal if the sample size were 𝑛 = 8?

17

( 2 ) In mountain country, major highways sometimes use tunnels instead of long, winding roads over high

passes. However, too many vehicles in a tunnel at a time can cause a hazardous situation. Traffic

engineers are studying a long tunnel in Colorado. If 𝑥 represents the time for a vehicle to go through the

tunnel, it is known that the 𝑥 distribution has a mean 𝜇 = 12.1 minutes and standard deviation 𝜎 = 3.8

minutes under ordinary traffic conditions. From a histogram of 𝑥 values, it was found that the 𝑥

distribution is mound-shaped with some symmetry about the mean.

Engineers have calculated that, on average, vehicle should spend from 11 to 13 minutes in the tunnel. If

the time is less than 11 minutes, traffic is moving too fast for safe travel in the tunnel. If the time is more

than 13 minutes, there is a problem of bad air quality (too much carbon monoxide and other pollutants).

Under ordinary conditions, there are about 50 vehicles in the tunnel at one time. What is the probability

that the mean time for 50 vehicles in the tunnel will be from 11 to 13 minutes?

a. Let �̅� represent the sample mean based on samples of size 50. Describe the �̅� distribution.

b. Find 𝑃(11 < �̅� < 13).

c. Interpret your answer to part (b).

18

Focus Point:

State assumptions

needed to use the

normal

approximation to the

binomial distribution

Compute 𝜇 and 𝜎 for

the normal

approximation

Use the continuity

correction to convert

a range of 𝑟 values to

normal 𝑥 values,

then into

standardized 𝑧 scores

to find desired

probabilities

Section 7.6 Normal Approximation to Binomial Distribution and to �̂� Distribution

Objective: In this lesson you learned how to compute 𝜇 and 𝜎 for the normal approximation; use

continuity correction to convert a range of 𝑟 values to a corresponding range of normal

𝑥 values; convert the 𝑥 values to a range of standardized 𝑧 scores and find desired

probabilities; and describe the sampling distribution for proportions �̂�.

I. Normal Approximations

What does a Normal Approximation to the Binomial tell us?

Normal Approximation to the Binomial Distribution

Consider a binomial distribution where

How to make a Continuity Correction

19

Focus Point:

Describe the

sampling distribution

for proportions �̂�

II. Sampling Distributions for Proportions

What does a �̂� Sampling Distribution tell us?

Sampling Distribution for the Proportion �̂� =𝒓

𝒏

20

Section 7.6 Examples – Normal Approximation to Binomial Distribution and to �̂�

Distribution

( 1 ) From many years of observation, a biologist knows that the probability is only 0.65 than any given Arctic

tern (a type of bird) will survive the migration from its summer nesting area to its winter feeding

grounds. A random sample of 500 Arctic terns were banded at their summer nesting area. Use the

normal approximation to the binomial and the following steps to find the probability that between 310

and 340 of the banded Arctic terns will survive the migration. Let 𝑟 be the number of surviving terns.

a. To approximate 𝑃(310 ≤ 𝑟 ≤ 340), we use the normal curve with 𝜇 = _________ and

𝜎 = _________.

b. 𝑃(310 ≤ 𝑟 ≤ 340) is approximately equal to 𝑃( _________ ≤ 𝑥 ≤ _________ ), where 𝑥 is a

variable from the normal distribution described in part (a).

c. Convert the condition 309.5 ≤ 𝑥 ≤ 340.5 to a condition in standard units.

d. 𝑃(310 ≤ 𝑟 ≤ 340) = 𝑃(309.5 ≤ 𝑥 ≤ 340.5) = 𝑃(−1.45 ≤ 𝑧 ≤ 1.45) = _________.

e. Will the normal distribution make a good approximation to the binomial for the problem?

Explain your answer.