SAMPLING DISTRIBUTION

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

- 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 lecture topic

p(X > 50) = ? frequency raw score 70 Distribution of Individuals in a Population

frequency raw score 70 p(X > 50) = 1 9 = 0.11 Distribution of Individuals in a Population

p(X > 30) = ? frequency raw score 70 Distribution of Individuals in a Population

p(X > 30) = 6 9 = frequency raw score 70 Distribution of Individuals in a Population

frequency normally distributed  = 40,  = 10 Distribution of Individuals in a Population p(40 < X < 60) = ?

frequency normally distributed  = 40,  = 10 p(40 < X < 60) = p(0 < Z < 2) = Distribution of Individuals in a Population

frequency normally distributed  = 40,  = 10 raw score Distribution of Individuals in a Population p(X > 60) = ?

frequency raw score 70 normally distributed  = 40,  = 10 p(X > 60) = p(Z > 2) = Distribution of Individuals in a Population

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 frequency raw score 7 8 9

Distribution of Sample Means from Samples of Size n = 2 12, 22 22,43 32,64 42,85 54,23 64,44 74,65 84,86 96,24 106,45 116,66 126,87 138,25 148, Sample # Scores Mean ( )

Distribution of Sample Means from Samples of Size n = frequency 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 = frequency sample mean p( > 7) = ?

Distribution of Sample Means from Samples of Size n = frequency sample mean

frequency raw score Distribution of Individuals in Population Distribution of Sample Means frequency sample mean  = 5,  = 2.24  X = 5,  X = 1.58

frequency raw score frequency sample mean Distribution of Individuals Distribution of Sample Means  = 5,  = 2.24 p(X > 7) = 0.25  X = 5,  X = 1.58 p(X> 7) = , for n=2

What if we took a larger sample?

Distribution of Sample Means from Samples of Size n = frequency sample mean =  X = 5,  X = 1.29 p( X > 7) = p(X > 7) = 0.25 p(X> 7) = , for n=2 for n = 3

frequency sample mean Distribution of Sample Means Things to Notice 1.The sample means tend to pile up around the population mean (stays centered at the population mean). 2.The distribution of sample means is approximately normal in shape, even though the population distribution was not (becomes more normal). 3.The distribution of sample means has less variability than does the population distribution (becomes less variable).

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

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 (Selisihan Piawai) SE =

frequency raw score Distribution of Individuals in Population Distribution of Sample Means frequency sample mean  = 5,  = 2.24  X = 5,  X = 1.58

frequency sample mean Sampling Distribution (n = 3)  X = 5  X = 1.29

Consider a finite population of 4000 students of a certain college. Of this 200 students are selected randomly and their mean weight is calculated. It was computed as x-bar = 135 lbs. If we draw 20 different samples of 200 students, instead of just one as we did previously. We can’t expect the same mean weight of these 20 samples chosen randomly. It is due to sampling variability i.e. each sample drawn randomly from the same population differs from each other in their computed 'statistic'.

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”.

PopulationSample Distribution of Sample Means Clarifying Formulas notice

The Central Limit Theorem Regardless the shape of the underlying population distribution, the sampling distribution of the mean becomes approximately normal as the sample size increase. Thus -if the random variable X is normally distributed, then the sampling distribution of the sample mean is normally distributed also regardless the size of the sample -for all other random variables X, the sampling distributions are approximately normally distributed if the size of the sample is large enough (n≥30) 1.will have a mean of  2.will have a standard error of 3.will approach a normal distribution as n approaches infinity What does this mean in practice?

The Central Limit Theorem Applies to Sampling Distribution of Any Population

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 nlarge n normal population non-normal population

Applications of the Central Limit Theorem The primary use of the distribution of sample means is to find the probability associated with any specific sample using the standard normal distribution. Population distribution Sampling distribution

Given the population of women has normally distributed weights with a mean of 64 kg and a standard deviation of 4 kg, 1.if one woman is randomly selected, find the probability that her weight is greater than 68 kg. 2.if 16 women are randomly selected, find the probability that their mean weight is greater than 68 kg. Applications of the Central Limit Theorem

0 Given the population of women has normally distributed weights with a mean of 64 kg and a standard deviation of 4 kg, 1.if one woman is randomly selected, find the probability that her weight is greater than 68 kg. 68  = 64 Population distribution z =

0 2.if 16 women are randomly selected, find the probability that their mean weight is greater than 68 kg. 68  = 64 Sampling distribution Given the population of women has normally distributed weights with a mean of 64 kg and a standard deviation of 4 kg,

Given the population of women has normally distributed weights with a mean of 64kg and a standard deviation of 4 kg 1.if one woman is randomly selected, find the probability that her weight is greater than 68 kg. 2.if 16 women are randomly selected, find the probability that their mean weight is greater than 68 kg. Applications of the Central Limit Theorem

Practice Given a population of 4000 academic books with a mean price = RM 50, and a standard deviation = RM 6, 1.What is the standard error of the sample mean for a sample of size 1? 2.What is the standard error of the sample mean for a sample of size 4? 3.What is the standard error of the sample mean for a sample of size 25? 4.What is the standard error of the sample mean for a sample of size 36? 5.Based on parts 1, 2, 3 and 4, what can we conclude about standard error?

1.if one book is randomly selected from the population, find the probability that its price is greater than RM If 4 books are randomly selected from the population, find the probability that their mean price is greater than RM 53 3.If 25 books are randomly selected from the population, find the probability that their mean price is greater than RM 53 Practice Given a population of 4000 academic books, with a mean price = RM 50, and a standard deviation = RM 6,