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Probability and Samples: The Distribution of Sample Means - larger sample…
Probability and Samples: The Distribution of Sample Means - larger sample size has a better probability outcome.
Sampling Error - the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.
Samples are variables, they are not always the same size.
The general concept of sampling error is that a sample typically will not provide a perfectly accurate representation of its population. More specifically, there typically is some discrepancy (or error) between a statistic computed for a sample and the corresponding parameter for the population.
Sampling Distribution - a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.
Distribution of Sample Means - the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population.
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The pile of sample means should tend to form a normal-shaped distribution. Logically, most of the samples should have means close to μ, and it should be relatively rare to find sample means that are substantially different from μ. As a result, the sample means should pile up in the center of the distribution (around μ) and the frequencies should taper off as the distance between M and μ increases. This describes a normal-shaped distribution.
In general, the larger the sample size, the closer the sample means should be to the population mean, μ.Logically, a large sample should be better than a small sample because it is more representative. Thus, the sample means obtained with a large sample size should cluster relatively close to the population mean; the means obtained from small samples should be more widely scattered.
The sample means should pile up around the population mean. Samples are not expected to be perfect but they are representative of the population. As a result, most of the sample means should be relatively close to the population mean.
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/square root of n) and will approach a normal distribution as n approaches infinity.
describes the distribution of sample means for any population, no matter what shape, mean, or standard deviation.
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Expected Value of M - the mean of the distribution of sample means always is identical to the population mean.
Sample mean is an example of an unbiased statistic, which means that on average the sample statistic produces a value that is exactly equal to the corresponding population parameter.
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First, we have the original population of scores. This population contains the scores for thousands or millions of individual people, and it has its own shape, mean, and standard deviation.
Next, we have a sample that is selected from the population. The sample consists of a small set of scores for people who have been selected to represent the entire population.
The third distribution is the distribution of sample means. This is a theoretical distribution consisting of the sample means obtained from all the possible random samples of a specific size.
The primary use for the distribution of sample means is to find the probability of selecting a sample with a specific mean
The z-score tells exactly where the sample mean is located in relation to all the other possible sample means that could have been obtained.
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The number tells the distance between the location and the mean in terms of the number of standard deviations.
However, we are now finding a location within the distribution of sample means. Therefore, we must use the notation and terminology appropriate for this distribution. First, we are finding the location for a sample mean (M) rather than a score (X). Second, the standard deviation for the distribution of sample means is the standard error, σM.
The sign tells whether the sample mean is located above (+) or below (−) the mean for the distribution (which is the population mean, μ.
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