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Lecture

# Chapter 9

17 Pages
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Department
Quantitative Methods
Course Code
QMS 202
Professor
Jason Chin- Tiong Chan

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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
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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Description
QMS202-Business Statistics II Chapter9 Outcomes: 1. Review 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 Page1 www.notesolution.com 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 , divided by the population standard deviation . The formula is 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? x 3000 2500 z = = = 1.25 400 Area Under the Normal Curve 1. Approximately 68 percent of the area under the normal curve is within one standard deviation of the mean 2. Approximately 95 percent of the area under the normal curve is within two standard deviations of the mean 2 3. Approximately 99.7 percent of the area under the normal curve is within three standard deviations of the mean 3 Winter2011 Page2 www.notesolution.com 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 x= and x = n The sample size is usually considered to be large ifn 30 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 = x Winter2011 Page3 www.notesolution.com QMS202-Business Statistics II Chapter9 Standard Error of the Mean If the standard deviation of the population is , the standard deviation of the distribution of the sample mean is x = n Example3 The mean rent for all two-bedroom apartments in downtown Toronto is \$1250 with a standard deviation of \$325. However the population distribution of rents for two-bedroom in downtown Toronto is skewed to the right. Determine the mean and standard deviation of x and describe the shape of its sampling distribution when the sample size is a. 36 b. 100 Sampling from a Normal Population If a population follows the normal distribution, the sampling distribution of the sample mean will also follow the normal distribution Finding the z value ofx when is known 1. Given that the standard deviation of the population, is known 2. The z value is x z = n Finding the z value ofx when is unknown If the population standard deviation of the population, is unknown and n 30 , the z value is z = x s n Winter2011 Page4 www.notesolution.com
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