Probability and Samples Sampling Distributions Central Limit Theorem Standard Error Probability of Sample Means

Inferential Statistics tomorrow and beyond Population Sample Probability last week and today

- getting a certain type of individual when we sample once When we take a sample from a population we can talk about the probability of - getting a certain type of individual when we sample once today last Thursday - getting a certain type of sample mean when n>1

Distribution of Individuals in a Population 10 20 30 40 50 60 1 2 3 frequency 4 5 6 raw score 70 p(X > 50) = ?

Distribution of Individuals in a Population 10 20 30 40 50 60 1 2 3 frequency 4 5 6 raw score 70 1 p(X > 50) = = 0.11 9

Distribution of Individuals in a Population 10 20 30 40 50 60 1 2 3 frequency 4 5 6 raw score 70 p(X > 30) = ?

Distribution of Individuals in a Population 10 20 30 40 50 60 1 2 3 frequency 4 5 6 raw score 70 6 p(X > 30) = = 0.66 9

Distribution of Individuals in a Population 6 normally distributed  = 40,  = 10 5 frequency 4 3 2 1 10 20 30 40 50 60 70 p(40 < X < 60) = ?

Distribution of Individuals in a Population 6 normally distributed  = 40,  = 10 5 frequency 4 3 2 1 10 20 30 40 50 60 70 p(40 < X < 60) = p(0 < Z < 2) = 47.7%

Distribution of Individuals in a Population 6 normally distributed  = 40,  = 10 5 frequency 4 3 2 raw score 1 10 20 30 40 50 60 70 p(X > 60) = ?

Distribution of Individuals in a Population 6 normally distributed  = 40,  = 10 5 frequency 4 3 2 raw score 1 10 20 30 40 50 60 70 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: Each individual in the population has an equal chance of being selected. If more than one individual is to be selected, there must be constant probability for each and every selection (i.e. sampling with replacement).

Distribution of Sample Means 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.

Population 6 5 frequency 4 3 2 1 raw score 1 2 3 4 5 6 7 8 9

Distribution of Sample Means from Samples of Size n = 2 Sample # Scores Mean ( ) 1 2, 2 2 2,4 3 2,6 4 2,8 5 4,2 6 4,4 7 4,6 8 4,8 9 6,2 10 6,4 11 6,6 12 6,8 13 8,2 14 8,4 15 8.6 16 8.8

Distribution of Sample Means from Samples of Size n = 2 6 5 frequency 4 3 2 1 1 2 3 4 5 6 7 8 9 sample mean We can use the distribution of sample means to answer probability questions about sample means

Distribution of Sample Means from Samples of Size n = 2 6 5 frequency 4 3 2 1 1 2 3 4 5 6 7 8 9 sample mean p( > 7) = ?

Distribution of Sample Means from Samples of Size n = 2 6 5 frequency 4 3 2 1 1 2 3 4 5 6 7 8 9 sample mean 1 p( > 7) = = 6 % 16

Distribution of Individuals in Population 1 2 3 4 5 6 frequency raw score 7 8 9  = 5,  = 2.24 X = 5, X = 1.58 Distribution of Sample Means 6 5 frequency 4 3 2 1 1 2 3 4 5 6 7 8 9 sample mean

Distribution of Individuals 1 2 3 4 5 6 frequency raw score 7 8 9  = 5,  = 2.24 Distribution of Sample Means 6 X = 5, X = 1.58 5 frequency 4 3 2 p(X > 7) = 25% 1 1 2 3 4 5 6 7 8 9 sample mean 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”.

Distribution of Individuals in Population 1 2 3 4 5 6 frequency raw score 7 8 9  = 5,  = 2.24 X = 5, X = 1.58 Distribution of Sample Means 6 5 frequency 4 3 2 1 1 2 3 4 5 6 7 8 9 sample mean

Things to Notice Distribution of Sample Means frequency 1 2 3 4 5 6 frequency 7 8 9 sample mean Distribution of Sample Means Things to Notice The sample means tend to pile up around the population mean. The distribution of sample means is approximately normal in shape, even though the population distribution was not. The distribution of sample means has less variability than does the population distribution.

What if we took a larger sample?

Distribution of Sample Means from Samples of Size n = 3 24 22 20 X = 5, X = 1.29 18 frequency 16 14 1 64 = 2 % p( X > 7) = 12 10 8 6 4 2 1 2 3 4 5 6 7 8 9 sample mean

Distribution of Sample Means As the sample gets bigger, the sampling distribution… stays centered at the population mean. becomes less variable. becomes more normal.

Central Limit Theorem For any population with mean  and standard deviation , the distribution of sample means for sample size n … will have a mean of  will have a standard deviation of will approach a normal distribution as n approaches infinity

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

Standard Error 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. SE =

Standard Error The Law of Large Numbers states: The larger the sample size, the smaller the standard error. This makes sense from the formula for standard error …

Distribution of Individuals in Population 1 2 3 4 5 6 frequency raw score 7 8 9  = 5,  = 2.24 X = 5, X = 1.58 Distribution of Sample Means 6 5 frequency 4 3 2 1 1 2 3 4 5 6 7 8 9 sample mean

Sampling Distribution (n = 3) 1 2 3 4 5 6 frequency 8 10 12 7 9 sample mean 14 16 18 20 22 24 X = 5 X = 1.29

Clarifying Formulas Distribution of Sample Means Population Sample notice

will have a standard deviation of Central Limit Theorem For any population with mean  and standard deviation , the distribution of sample means for sample size n … will have a mean of  will have a standard deviation of will approach a normal distribution as n approaches infinity 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

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.

Probability and the Distribution of Sample Means Example: Given the population of women has normally distributed weights with a mean of 143 lbs and a standard deviation of 29 lbs, if one woman is randomly selected, find the probability that her weight is greater than 150 lbs. if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lbs.

Given the population of women has normally distributed weights with a mean of 143 lbs and a standard deviation of 29 lbs, if one woman is randomly selected, find the probability that her weight is greater than 150 lbs. 150  = 143 = 29 Population distribution z = 150-143 = 0.24 29 0.4052 0.24

Given the population of women has normally distributed weights with a mean of 143 lbs and a standard deviation of 29 lbs, if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lbs. 150  = 143 = 4.33 Sampling distribution z = 150-143 = 1.45 4.33 0.0735 1.45

Probability and the Distribution of Sample Means Example: Given the population of women has normally distributed weights with a mean of 143 lbs and a standard deviation of 29 lbs, if one woman is randomly selected, find the probability that her weight is greater than 150 lbs. if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lbs.

Practice Given a population of 400 automobile models, with a mean horsepower = 105 HP, and a standard deviation = 40 HP, Example: What is the standard error of the sample mean for a sample of size 1? What is the standard error of the sample mean for a sample of size 4? What is the standard error of the sample mean for a sample of size 25? 40 20 8

Practice Example: Given a population of 400 automobile models, with a mean horsepower = 105 HP, and a standard deviation = 40 HP, if one model is randomly selected from the population, find the probability that its horsepower is greater than 120. If 4 models are randomly selected from the population, find the probability that their mean horsepower is greater than 120 If 25 models are randomly selected from the population, find the probability that their mean horsepower is greater than 120 .35 .23 .03