School

Ryerson University
Department

Quantitative Methods

Course Code

QMS 202

Professor

Jason Chin- Tiong Chan

QMS202-Business Statistics II Chapter9

Outcomes:

1.Review â€“ the standardized normal random variable z and the sampling

distribution

2.Construct and interpret confidence interval for the population mean

when the population standard deviation is known.

3.Construct and interpret confidence interval for the population mean

when the population standard deviation is unknown.

4.Construct a confidence interval for a population proportion.

5.Determine the sample size necessary to develop a confidence interval

for the mean or proportion.

Review

Characteristics of a Normal Probability Distribution

1. It is bell-shaped and has a single peak at the exact centre of the

distribution

2.The arithmetic mean, median, and mode are equal and located at the

peak

3.Half the area under the curve is above the mean and half is below the

mean

4.The normal probability distribution is symmetrical about its mean

5.The normal probability distribution is asymptotic. That is the curve

gets closer and closer to the X-axis on each side, but never actually

touches it.

Winter2011 Page#1

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QMS202-Business Statistics II Chapter9

The Standard Normal Probability Distribution

1.The standard normal distribution has mean of 0 and standard

deviation of 1. It is also called the z distribution.

2. A z-value or standard normal value is the distance between a selected

value, designed x and the population mean

Âµ

, divided by the

population standard deviation

Ïƒ

. The formula is

Ïƒ

Âµ

âˆ’

=

x

z

Example1

The bi-weekly starting salaries of recent MBA graduates from Ryerson

University follows the normal distribution with a mean of $2,500 and a

standard deviation of $400. What is the z-value for a salary of $3000?

Ïƒ

Âµ

âˆ’

=

x

z

=

400

25003000

âˆ’

= 1.25

Area Under the Normal Curve

1.Approximately 68 percent of the area under the normal curve is within

one standard deviation of the mean

ÏƒÂµ

Â±

2.Approximately 95 percent of the area under the normal curve is within

two standard deviations of the mean

ÏƒÂµ

2

Â±

3.Approximately 99.7 percent of the area under the normal curve is

within three standard deviations of the mean

ÏƒÂµ

3

Â±

Winter2011 Page#2

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QMS202-Business Statistics II Chapter9

Note: This is the Empirical Rule

Example2

Assume that the annual earnings of all employees with Certified

Management Accountant (CMA) certifications, 10 years of experience and

working for large firms have a bell-shaped distribution with a mean of

$89,500 and a standard deviation of $13,400.

a. Approximately 68 percentage of all such employees have annual earnings

between __76,100__ and __102.900__ .

b. Approximately 95 percentage of all such employees have annual earnings

between __62,700__ and __116,300__ .

c. Approximately 99.7 percentage of all such employees have annual

earnings between __49,300__ and __129,700__ .

Why sample the population

1.To contact the whole population would often be time prohibitive

2.The cost of studying all the items in the population may be prohibitive

3.The sample results are adequate

4.The destructive nature of some tests. Example If the wine tasters in

Niagara-on-the-Lake drank all the wine to evaluate the vintage, they

would consume the entire crop, and none would be available for sale.

5.The physical impossibility of checking all items in the population

Central Limit Theorem

For a large sample size, the sampling distribution of

x

is approximately

normal, irrespective of the shape of the population distribution. The mean

and standard deviation of the sampling distribution of

x

are

ÂµÂµ

=

x

and

n

x

Ïƒ

Ïƒ

=

The sample size is usually considered to be large if

30

â‰¥

n

Mean of the Sample Means

The mean of the distribution of the sample mean will be exactly equal to the

population mean if we are able to select all possible samples of a particular

size from a given population

ÂµÂµ

=

x

Winter2011 Page#3

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