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Probability and Samples, Construct the Distribution of Sample Means -…
Probability and Samples
Sampling Error: The natural difference between a sample statistic and its corresponding population parameter.
Sample Variability: The natural difference between samples taken from the same population; two separate samples from the same population will not be exactly the same.
Distribution of Sample Means: A distribution of the means from all possible samples of the same size selected from a population.
Distribution of Sample Means: Select many random samples of the same size (n) from a population, calculate the mean of each sample, and place the means into a frequency distribution. Repeating this process creates the distribution of sample means.
The process of selecting samples of n scores, calculating each sample mean, and placing the means into a frequency distribution until all possible sample means are included.
Characteristics of the Distribution of Sample Means: The distribution describes the center, spread, and shape of sample means from all possible samples of the same size.
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- Center: Sample means tend to cluster around the population mean (μ).
- Shape: The distribution of sample means tends to have a normal shape, with most means near μ and fewer means farther away.
- Sample Size: Larger samples produce sample means that are closer to the population mean and less spread out than smaller samples.
Mean of the Distribution of Sample Means: The mean of the distribution of sample means is represented by μ and is equal to the population mean μ.
Shape, Central Tendency, and Variability for the Distribution of Sample Means
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Variability: The sample means have a standard error, which measures how much the sample means vary around the population mean.
Central Limit Theorem: A theorem stating that the distribution of sample means from all possible samples tends to be normal, with a mean equal to the population mean (μ) and a standard error that decreases as the sample size increases.
Shape of the Distribution of Sample Means: The distribution of sample means tends to be normal, especially when the population is normally distributed or when the sample size is large (about 30 or more).
As the sample size (n) gets larger, the distribution of sample means becomes more like a normal distribution, regardless of the shape of the original population
Mean of the Distribution of Sample Means: The expected value of M is equal to the population mean (μ).
Expected Value of M: The average of all possible sample means is equal to the population mean (μ). In other words, a sample mean is expected to be close to the population mean.
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Standard Error of M: The standard deviation of the distribution of sample means. It measures how much sample means vary from the population mean.
Law of Large Numbers: As sample size increases, the sample mean tends to get closer to the population mean, making the sample more accurate.
Z-Scores and Probability for Sample Means: A z-score shows how far a sample mean is from the population mean in terms of standard errors and can be used to find the probability of obtaining a particular sample mean.
Sampling Error and Standard Error: Sampling error is the difference between a sample mean and the population mean, while standard error measures how much sample means vary around the population mean.
Inferential Statistics: Methods that use sample data to make general conclusions about a population. Because a sample mean may differ from the population mean, the standard error of M measures the average amount of difference expected between the sample mean and the population mean.
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