Download 6.5 Day 2: The Central Limit Theorem

6.39 Day 2: The Central
Limit Theorem
By the end of class you will be
able to calculate probabilities
involving sample means
A problem we know…
• The average teacher’s salary in North
Dakota is $29,863. Assume a normal
distribution with a standard deviation of
$5,100.
• What is the probability that a random
teacher’s salary is more than $40,000?
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A new problem…Sample Means
• The average teacher’s salary in North
Dakota is $29,863. Assume a normal
distribution with a standard deviation of
$5,100.
• What is the probability that the mean for
a sample of 80 teachers’ salaries is
greater than $30,000?
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The Central Limit Theorem states
when we can use a normal
distribution to solve problems….
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Central Limit Theorem
• As the sample size n increases without limit, the
shape of the distribution of the sample means
taken with replacement from a population with
mean  and standard deviation  will
approach a normal distribution with mean  and
standard deviation n
z
ALSO The Standard
Score Must Change:
x

n
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In English….
• If the sample size is larger than 30, a
distribution of sample means can be
approximated using the normal distribution
• If the original population is normally
distributed, then the sample means will be
normally distributed for any sample size
• As the sample size increases, the
sampling distribution of sample means
approaches a normal distribution
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Sampling Distribution of Sample
Means
• Distribution formed by using the means
computed from all possible random
samples of a specific size taken from a
population.
• Sampling Error: The difference between
the sample measure and the
corresponding population measure since
samples are not perfect representations of
populations.
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A new problem…Sample Means
• The average teacher’s salary in North
Dakota is $29,863. The standard deviation
is $5,100.
• What is the probability that the mean for
a sample of 80 teachers’ salaries is
greater than $30,000?
8
At a proofreading company the
average age of employees is 36.2
years and the standard deviation is
3.7 years.
• If a person from the company is chosen at
random, find the probability that her age
will be between 36 and 37.5 years.
Assume the variable is normally
distributed.
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At a proofreading company the
average age of employees is 36.2
years and the standard deviation is
3.7 years.
• If a random sample of 15 proofreaders is
selected find the probability that the mean
age of proofreaders in the sample will be
between 36 and 37.5 years. Assume the
original distribution is normally distributed.
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Summarizer:
• The most important thing about the
distribution of sample means is……
• I must also remember …..
• And……
• But the most important thing to remember
about the distribution of sample means
is…
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