probability and samples sampling distributions central limit theorem standard error probability of...

Probability and Samples

• Sampling Distributions

• Central Limit Theorem

• Standard Error

• Probability of Sample Means

Sample Population

Inferential Statistics

Probability

last week and today

tomorrow and beyond

- getting a certain type of individual when we sample once

- getting a certain type of sample mean when n>1

When we take a sample from a population we can talk about the probability of

today

last Thursday

p(X > 50) = ?

10 20 30 40 50 60

1

2

3

freq

uenc

y4

5

6

raw score

70

Distribution of Individuals in a Population

10 20 30 40 50 60

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70

p(X > 50) = 1 9

= 0.11


p(X > 30) = ?

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70


p(X > 30) = 6 9

= 0.66

10 20 30 40 50 60

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raw score

70


10 20 30 40 50 60

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3freq

uenc

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70

normally distributed = 40, = 10


p(40 < X < 60) = ?

10 20 30 40 50 60

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70


p(40 < X < 60) = p(0 < Z < 2) = 47.7%


10 20 30 40 50 60

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70


raw score


p(X > 60) = ?

10 20 30 40 50 60

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raw score70


p(X > 60) = p(Z > 2) = 2.3%


For the preceding calculations to be accurate, it is necessary that the sampling process be random.

A random sample must satisfy two requirements:

1. Each individual in the population has an equal chance of being selected.

2. If more than one individual is to be selected, there must be constant probability for each and every selection (i.e. sampling with replacement).

A distribution of sample means is:

the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population.

Distribution of Sample Means

Population

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raw score

7 8 9

Distribution of Sample Means from Samples of Size n = 2

1 2, 2 2

2 2,4 3

3 2,6 4

4 2,8 5

5 4,2 3

6 4,4 4

7 4,6 5

8 4,8 6

9 6,2 4

10 6,4 5

11 6,6 6

12 6,8 7

13 8,2 5

14 8,4 6

15 8.6 7

16 8.8 8

Sample # Scores Mean ( )

X


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sample mean

We can use the distribution of sample means to answer probability questions about sample means


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sample mean

p( > 7) = ?

X


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7 8 9

sample mean

p( > 7) = 116

= 6 %

X

1 2 3 4 5 6

123fr

eque

ncy

456

raw score7 8 9

Distribution of Individuals in Population


1 2 3 4 5 6

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7 8 9

sample mean

= 5, = 2.24

X = 5, X = 1.58

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123fr

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raw score7 8 9

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sample mean

Distribution of Individuals


= 5, = 2.24

p(X > 7) = 25%

X = 5, X = 1.58

p(X> 7) = 6% , for n=2

A key distinction

Population Distribution – distribution of all individual scores in the population

Sample Distribution – distribution of all the scores in your sample

Sampling Distribution – distribution of all the possible sample means when taking samples of size n from the population. Also called “the distribution of sample means”.

1 2 3 4 5 6

123fr

eque

ncy

456

raw score7 8 9



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7 8 9

sample mean

= 5, = 2.24

X = 5, X = 1.58

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sample mean


Things to Notice

1. The sample means tend to pile up around the population mean.

2. The distribution of sample means is approximately normal in shape, even though the population distribution was not.

3. The distribution of sample means has less variability than does the population distribution.

What if we took a larger sample?


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7 8 9

sample mean

14

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24

164

= 2 %

X = 5, X = 1.29

p( X > 7) =


As the sample gets bigger, the sampling distribution…

1. stays centered at the population mean.

2. becomes less variable.

3. becomes more normal.

Central Limit Theorem

For any population with mean and standard deviation , the distribution of sample means for sample size n …

1. will have a mean of

2. will have a standard deviation of

3. will approach a normal distribution as n approaches infinity

n

Notation

the mean of the sampling distribution

the standard deviation of sampling distribution (“standard error of the mean”)

X

nX

The “standard error” of the mean is:

The standard deviation of the distribution of sample means.

The standard error measures the standard amount of difference between x-bar and that is reasonable to expect simply by chance.

Standard Error

SE =

n

The Law of Large Numbers states:

The larger the sample size, the smaller the standard error.

Standard Error

This makes sense from the formula for standard error …

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123fr

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raw score7 8 9



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sample mean

= 5, = 2.24

X = 5, X = 1.58

58.12

24.2X

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7 8 9

sample mean

14

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24

Sampling Distribution (n = 3)

X = 5X = 1.29

29.13

24.2X

Population SampleDistribution of Sample Means

Clarifying Formulas

N

X n

XX X

N

ss

1

n

sss nX

nX

22

notice

Central Limit Theorem

For any population with mean and standard deviation , the distribution of sample means for sample size n …

1. will have a mean of

2. will have a standard deviation of

3. will approach a normal distribution as n approaches infinity

n

What does this mean in practice?

Practical Rules Commonly Used:

1. For samples of size n larger than 30, the distribution of the sample means can be approximated reasonably well by a normal distribution.

The approximation gets better as the sample size n becomes larger.

2. If the original population is itself normally distributed, then the sample means will be normally distributed for any sample size.

small n large n

normal population

non-normal population

normal is X normal is X

normal is Xnonnormal is X

Probability and the Distribution of Sample Means

The primary use of the distribution of sample means is to find the probability associated with any specific sample.


Given the population of women has normally distributed weights with a mean of 143 lbs and a standard deviation of 29 lbs,

Example:

1. if one woman is randomly selected, find the probability that her weight is greater than 150 lbs.

2. if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lbs.

0 0.24



0.4052

150 = 143

= 29

Population distribution

z = 150-143 = 0.24 29

0 1.45


0.0735


36

29X

150 = 143

= 4.33

Sampling distribution

z = 150-143 = 1.45 4.33



Example:



41.)150( XP

07.)150( XP

Practice

Given a population of 400 automobile models, with a mean horsepower = 105 HP, and a standard deviation = 40 HP,

Example:

1. What is the standard error of the sample mean for a sample of size 1?



40

20

8

Example:

1. if one model is randomly selected from the population, find the probability that its horsepower is greater than 120.

2. If 4 models are randomly selected from the population, find the probability that their mean horsepower is greater than 120

3. If 25 models are randomly selected from the population, find the probability that their mean horsepower is greater than 120

Practice

Given a population of 400 automobile models, with a mean horsepower = 105 HP, and a standard deviation = 40 HP,

.35

.23

.03

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