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Math
Statistics
Hypothesis Testing
Statistics Midterm: Distribution of Sample Means and Null testing Lecture 3
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What is the distribution of sample means?
the distribution of means obtained from all possible random samples of a particular size that could be drawn from a population
Previously, values in the distribution have been individual scores
With the distribution of sample means, observations in the distribution are SAMPLE MEANS
In practice, (functionally) countless samples could be drawn from a given population
Impossible to account for and calculate all possible samples directly
What are sampling distribution characteristics?
1. The sample means pile up around the population mean
-The "mean of the sample means" will be equal to the population mean
2. The distribution of sample means will generally form a normal distribution***
3. The distribution of sample means can be used to answer questions about the probability of selecting a random sample having a specific mean
What is the central limit theorem?
For actual populations, mathematical theory is used to describe the distribution of means for all possible samples of a given N
Central Limit Theorem
For any population with mean μ and standard deviation σ, the distribution of sample means for sample size N will have a mean of μ and a standard deviation of σ/√N and will approach a normal distribution as N approaches infinity
The distribution of sample means begins to approach a normal curve if either of the following conditions are met:
1. ) The population from which the samples are drawn is normal
2.) The sample N is around 30 or more
What will the sample mean be equal to in a distribution of sample means?
The mean of the distribution of sample means (i.e., the average of all possible sample means of a given N) will equal μ
Expected value of M:
refers to the mean of a distribution of sample means and is equal to the population μ
The standard error or mean:
the SD of the distribution of sample means
What does variablity in distribution of sample means tell us?
Provides an index of how much difference is expected from one sample to the next
Measures how well an individual sample mean estimates the population mean (i.e., error)
Indicates the expected difference between a sample mean and the population mean
What factors influence standard error in distribution of sample means?
1. Population standard deviation:
Standard error decreases with the population standard deviation
2. Sample size:
Law of large numbers: the larger the sample (N) the more likely it is that the sample mean will be close to the population mean
Distinguish the three distributions:
Distribution of POPULATION scores
Contains all available scores in a population
Has its own shape, mean, and variability (σ)
Distribution of SAMPLE scores
Contains a subset of scores selected to represent the population
Has its own shape, mean, and variability (s)
Distribution of SAMPLE MEANS
Contains the sample means from all possible samples of a specific size
Has its own shape, mean, and variability (σM)
What is the primary function of the distribution of sample means?
is to find the probability of randomly selecting a sample demonstrating certain characteristics, based on an assumed model
What is the probability of randomly selecting a sample with mean M, assuming a normal distribution of sample means?
Z scores for sample means
same methods used to locate probabilities associated with specific samples z= X-u/SD of M
How do you interpret patterns of data?
Several explanations for patterns of data:
1.) Pattern is caused by some systematic factor that reflects a true effect in the larger population
2.) Pattern is the result of chance fluctuations in the data and is not representative of an actual effect
3.) Pattern in the result of systematic bias in the methods and/or violations of the underlying statistical model
What can facilitate statistical inference?
Null hypothesis significance testing (NHST)
Goal is to differentiate population-based associations from random fluctuations in our sample data
-No statistical test for the identification of terrible methods
-Detection requires a critical evaluation of procedures and underlying assumptions!!!
What is a hypothesis test?
a statistical procedure that uses sample data to draw inferences about an unknown population value
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