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Chapter Seven

Introduction to Sampling Distributions

Section 2

The Central Limit Theorem

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Key Points 7.2

• For a normal distribution, use mu and sigma to construct the theoretical sampling distribution for the statistic x – bar

• For large samples, use sample estimates to construct a good approximate sampling distribution for the statistic x-bar

• Learn the statement and underlying mean of the central limit theorem well enough to explain it to a friend who is intelligent, but (unfortunately) does not know much about statistics

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Let x be a random variable with a normal distribution with mean and standard deviation . Let be the sample mean

corresponding to random samples of size n taken from

the distribution.

x

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Facts about sampling distribution of the mean:

• The distribution is a normal distribution.

• The mean of the distribution is (the same mean as the original distribution).

• The standard deviation of the distribution is (the standard deviation of the original distribution, divided by the square root of the sample size).

xx

xn

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We can use this theorem to draw conclusions about means of samples taken from normal

distributions.

If the original distribution is normal, then the sampling distribution will be normal.

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The Mean of the Sampling Distribution

x

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The mean of the sampling distribution is equal to the

mean of the original distribution.

x

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The Standard Deviation of the Sampling Distribution

x

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The standard deviation of the sampling distribution is equal to the

standard deviation of the original distribution divided by the square

root of the sample size.

nx

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The time it takes to drive between cities A and B is normally distributed

with a mean of 14 minutes and a standard deviation of 2.2 minutes.

• Find the probability that a trip between the cities takes more than 15 minutes.

• Find the probability that mean time of nine trips between the cities is more than 15 minutes.

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Mean = 14 minutes, standard deviation = 2.2 minutes

• Find the probability that a trip between the cities takes more than 15 minutes.

3264.06736.000.1)45.0(

45.02.2

1415

zP

z14 15

Find this area

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Mean = 14 minutes, standard deviation = 2.2 minutes

• Find the probability that mean time of nine trips between the cities is more than 15 minutes.

73.09

2.2

n

14

x

x

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Mean = 14 minutes, standard deviation = 2.2 minutes

• Find the probability that mean time of nine trips between the cities is more than 15 minutes.

0853.04147.05.0)37.1z(P

37.173.0

1415z

14 15

Find this area

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What if the Original Distribution Is Not Normal?

Use the Central Limit Theorem!

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MOVIE!

V01081a.rm

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Central Limit Theorem

If x has any distribution with mean and standard deviation , then the sample

mean based on a random sample of size n will have a distribution that

approaches the normal distribution (with mean and standard deviation

divided by the square root of n) as n increases without bound.

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How large should the sample size be to permit the

application of the Central Limit Theorem?

In most cases a sample size of

n = 30 or more assures that the distribution will be approximately normal and the theorem will apply.

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Central Limit Theorem

• For most x distributions, if we use a sample size of 30 or larger, the distribution will be approximately normal.

x

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Central Limit Theorem

• The mean of the sampling distribution is the same as the mean of the original distribution.

• The standard deviation of the sampling distribution is equal to the standard deviation of the original distribution divided by the square root of the sample size.

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Central Limit Theorem Formula

x

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Central Limit Theorem Formula

nx

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Central Limit Theorem Formula

n

x

xz

x

x

/

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Application of the Central Limit Theorem

Records indicate that the packages shipped by a certain trucking company have a mean weight of 510 pounds and a standard deviation of 90 pounds. One hundred packages are being shipped today. What is the probability that their mean weight will be:

a. more than 530 pounds?b. less than 500 pounds?c. between 495 and 515 pounds?

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Are we authorized to use the Normal Distribution?

Yes, we are attempting to draw conclusions about means of large samples.

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Applying the Central Limit Theorem

What is the probability that their mean weight will be more than 530 pounds?Consider the distribution of sample means:

P( x > 530): z = 530 – 510 = 20 = 2.22 9 9

P(z > 2.22) = _______

9100/90,510 xx

.0132

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Applying the Central Limit Theorem

What is the probability that their mean weight will be less than 500 pounds?

P( x < 500): z = 500 – 510 = –10 = – 1.11 9 9

P(z < – 1.11) = _______.1335

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Applying the Central Limit Theorem

What is the probability that their mean weight will be between 495 and 515 pounds?

P(495 < x < 515) :

for 495: z = 495 – 510 = 15 = 1.67 9 9

for 515: z = 515 – 510 = 5 = 0.56 9 9

P( 1.67 < z < 0.56) = _______ .6648

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